Topological character of hydrodynamic screening in suspensions of hard spheres: an example of universal phenomenon
aa r X i v : . [ c ond - m a t . s o f t ] A ug Topological character of hydrodynamicscreening in suspensions of hard spheres:an example of universal phenomenon
Ethan E. Ballard , Arkady L. Kholodenko
375 H.L.Hunter laboratories, Clemson University, Clemson, SC 29634-0973, USA
Abstract
Although in the case of polymer solutions the existence of hydrodynamic screening has been theoret-ically established some time ago, use of the same methods for suspensions of hard spheres thus far havefailed to produce similar results. In this work we reconsider this problem. Using superposition of topologi-cal and London-style qualitative arguments we prove the existence of screening in hard sphere suspensions.Even though some of these arguments were employed initially for treatments of superconductivity andsuperfluidity, we find analogs of these phenomena in nontraditional settings such as in colloidal suspen-sions, turbulence, magnetohydrodynamics, etc. In particular, in suspensions we demonstrate that thehydrodynamic screening is an exact analog of Meissner effect in superconductors. The extent of screeningdepends on the volume fraction of hard spheres. The zero volume fraction limit corresponds to the normalstate. The case of finite volume fractions-to the mixed state typical for superconductors of the secondkind with such a state becoming fully ”superconducting” at the critical volume fraction ϕ ∗ for which the(zero frequency) relative viscosity η ( relative ) diverges. Brady and, independently, Bicerano et al usingscaling-type arguments predicted that for ϕ close to ϕ ∗ the viscosity η ( relative ) behaves as C (1 − ϕ/ϕ ∗ ) − with C being some constant. Their prediction is well supported by experimental data. In this work weexplain such a behavior of viscosity in terms of a topological-type transition which mathematically can bemade isomorphic to the more familiar Bose-Einstein condensation transition. Because of this, the resultsand methods of this work are not limited to suspensions. In the concluding section we describe otherapplications ranging from turbulence and magnetohydrodynamics to high temperature superconductorsand QCD, etc. P ACS : 47.57. E-; 47.57.Qk; 47.65.-d; 66.20.Cy; 67.25.dk; 11.25.-w; 12.38.-t; 14.80.Hv; 47.10 Df
Keywords : Colloidal suspensions; Generalized Stokes-Einsten relation; Hydrodynamics and hydrody-namic screening; Bose-Einstein condensation; London equation; Ginzburg-Landau theory of superconduc-tivity; Topological field theories; Theory of knots and links; Helicity in hydro and magnetohydrodynamics;Classical mechanics in the vortex formalism; High T c superconductivity; Abelian projection for QCD;String models Sections 1-4 of this work comprise part of the thesis of E.Ballard done in compliance with requirements for the PhDdegree in materials science. Corresponding author
E-mail addresses : [email protected] (A.Kholodenko), [email protected] (E.Ballard) Introduction
In his 1905-1906 papers on Brownian motion for suspensions of hard spheres, Einstein obtained now fa-mous relation for the self-diffusion coefficient D for the noninteracting hard spheres of radius R immersedin a solvent at temperature T [1]: D = k B Tγ = k B T πRη . (1.1)In this formula η is the viscosity of a pure solvent and k B is the Boltzmann’s constant. This result isvalid only in the infinite dilution limit. In another paper [2] Einstein took into account the effects of finiteconcentrations and obtained the first nonvanishing correction to η for small but finite concentrations. Itis given by η/η = 1 + 2 . ϕ + O (cid:0) ϕ (cid:1) (1.2)with ϕ being the volume fraction ϕ := nV πR . In this formula n is the number of monodisperse hardspheres in the volume V . If we formally replace η by η in Eq. (1.1), the obtained result can be cautiouslyused as a definition for the cooperative diffusion coefficient D , i.e. D = k B T πRη . (1.3)Below, we use symbols D for the self-diffusion coefficient and D for the cooperative diffusion coefficient.By combining Eq.s (1.1) - (1.3), we also obtain: D/D = 1 − . ϕ + O (cid:0) ϕ (cid:1) . (1.4)Eq.(1.4) compares well with experimental results, e.g. those discussed in Ref.[3] . Numerous attemptshave been made to obtain results like Eq.(1.4) systematically. The above results are restricted by theobservation that Stoke’s formula for friction γ is applicable only for time scales longer than the charac-teristic relaxation time τ r of the solvent, τ r := ρR /πη ) , e.g. see [4]. In this formula ρ is the density ofpure solvent. This requirement provides the typical cut-off time scale, while the parameter R serves as atypical space cut-off for the problems we are going to study in this work.By analogy with the theory of nonideal gases, expansion Eq. (1.2) is referred to as a ”virial”. Unlikethe theory of nonideal gases, where the virial coefficients are known exactly to a very high order [5],values for coefficients in the virial expansion for η have been an active area of research to date even in thelow concentration regime. A considerable progress was made in obtaining closed form approximationsdescribing the rheological properties of suspensions of hard spheres in a broad range of concentrations[6-8]. Similar results for particles of other geometries are much less complete [7,9]. An extension of theseresults to solutions of polymers has taken place in parallel with these developments [10]. A noticeable ad-vancements have been made in our understanding of rheology of dilute and semidilute polymer solutionsfor fully flexible polymers and rigid rods. It should be noted, though, that polymers add further com-plexities because the connectivity of the polymer chain backbone plays an essential role in calculations ofrheological properties of polymer solutions. The effect of chain connectivity on viscoelastic properties ofpolymer solutions has been an object of extensive discussion, and many of theoretical difficulties encoun-tered in describing these solutions are shared by suspensions of hard spheres. In particular, it is known[11], that particles immersed in a viscous fluid affect the motion of each other both hydrodynamically andby direct interaction (hard core, etc). Since the motion of particles in a fluid is correlated, it contributesto the distribution of local velocities within the fluid. Behavior of many systems (e.g. those listed inSection 6) other than the hard sphere suspensions happens to be closely related or even isomorphic tothat noticed in suspensions. This observation makes study of suspensions important in many areas ofphysics, chemistry and biology. For reasons which will become apparent upon reading, in this work weshall mention only physical applications.In a polymer solutions when the polymer concentration ϕ increases, it is believed that the hydro-dynamic interactions become unimportant due to the effects of hydrodynamic screening [12]. To our The data on page 5 and in Table 2 of this reference support our conjecture. ϕ ∗ . Thisphenomenon has been observed experimentally and is well documented, e.g. see Refs. [14-18]. All thesereferences are concerned with changes in rheological properties of suspensions occurring with changes inconcentration ϕ. Using scaling-type arguments Brady, Ref.[19], and, independently, Bicerano et al , Ref.[20], had found that near ϕ = ϕ ∗ the relative viscosity η/η diverges as η/η = C (1 − ϕ/ϕ ∗ ) − with C being some constant. Furthermore, as it is shown by Bicerano et al , such analytical dependence of relativeviscosity on concentration ϕ actually works extremely well for all concentrations. In view of (1.3), it isreasonable to expect vanishing of D for ϕ → ϕ ∗ . This phenomenon was indeed observed in Ref.[21]. .Theoretically, the result for relative viscosity was obtained as result of a combined nontrivial use oftopological and combinatorial arguments. Such arguments can also be used, for instance, for descriptionof the onset of turbulence in fluids or gases. As described in Ref.[22], such a regime in these substancesis characterized by the sharp increase in the viscosity (just like in suspensions). According to Chorin,Ref.[22], Section 6.8, one can think about such an increase as analogous to processes which take place insuperfluid He when one goes in temperatures from below to above λ − transition, that is from the super-fluid to normal fluid state. Such a transition is believed to be associated with uninhibited proliferationof tangled vortices on any scale. In this work we demonstrate that Chorin’s conjecture is indeed correct.This interpretation is possible only if both topological and combinatorial arguments are rigorously andcarefully taken into account. Surprisingly, when this is done, the emerging description becomes isomorphicto that known for the Bose-Einstein condensation transition. Because of this, in addition to turbulence,in concluding section of this work we briefly discuss a number of apparently different physical systemswhose behavior under certain conditions resembles that found in colloidal supensions.The rest of this paper is organized as follows. In section 2, we introduce notations, discuss experimentaldata with help of previously found generalized Stokes-Einstein relation [23] and make conjectures abouthow these results should be interpreted in the case if hydrodynamic screening does exist. Some familiaritywith Ginzburg-Landau (G-L) theory of superconductivity is expected for proper understanding of thisand the following sections. In Section 3 we study in detail how many particle diffusion processes shouldbe affected if hydrodynamic interactions are taken into account. The major new results of this sectionare given in Section 3.3. where we rigorously demonstrate that account of hydrodynamic interactionscauses modification of Fick’s laws of diffusion in the same way as presence of electromagnetic field causesmodification of the Schrodinger’s equation for charged particles The gauge fields emerging in the modifiedFick’s equations are of zero curvature implying involvement of the Chern-Simons topological field theory.The following Section 4 considers in detail the implications of the results obtained in Section 3. The majornew result of this section is given in Section 4.4. where we adopted the logic of the ground breaking paperby London and London [24] in order to demonstrate the existence of hydrodynamic screening. Thus, thephenomenon of screening in suspensions is analogous to the Meissner effect in superconductors [25]. InSection 5 we follow the logic of Ginzburg-Landau paper [26] elaborating the work by London brothersand develop similar G-L-type theory for suspensions. The major new result of this section is presented inSection 5.5. in which by using combinatorial and topological methods we reproduce the scaling resultsby Brady [19] and Bicerano et al, Ref.[20]. In Section 6 we place the obtained results in a much broadercontext. It is done with help of two key concepts: helicity and force-free fields. They had been inuse for some time in areas such as magnetohydrodynamics, fluid, plasma and gas turbulence, classicalmechanics written in hydrodynamic formalism but not in superconductivity or colloidal suspensions, etc.3n this section we mention as well other uses of these concepts in disciplines such as high temperaturesuperconductivity, quantum chromodynamics, string theory, non-Abelian fluids, etc. The paper alsocontains three appendices which are made sufficiently self contained. They are not only very helpful inproviding details supporting the results of the main text but also of independentl interest. In 1976, Batchelor obtained the following general result for the cooperative diffusion coefficient [27]: D ( ϕ ) = K ( ϕ )6 πηr ϕ − ϕ (cid:18) ∂µ∂ϕ (cid:19) p,T , (2.1)where K ( ϕ ) is the sedimentation coefficient of the particles in suspension and µ is the chemical potential.Batchelor obtained for K ( ϕ ) the following result: K ( ϕ ) = 1 − . ϕ + O (cid:0) ϕ (cid:1) (2.2)so that (2.1) with thus obtained first order result for K ( ϕ ) can be used only for low concentrations. InRef.[3] an attempt was made to extend Batchelor’s results to higher concentrations. This was achievedin view of the fact that ϕ − ϕ (cid:18) ∂µ∂ϕ (cid:19) p,T = (cid:18) ∂ Π ∂n (cid:19) p,T , (2.3)where Π is the osmotic pressure. Use of this result in (2.1) produces: D ( ϕ ) = K ( ϕ )6 πηr (cid:18) ∂ Π ∂n (cid:19) p,T . (2.4)The Carnahan-Starling equation of state for hard spheres can be used to obtain the following result forcompressibility (cid:18) ∂ Π ∂n (cid:19) p,T = k B T h (1 + 2 ϕ ) + ( ϕ − ϕ i (1 − ϕ ) , (2.5)thus converting equation (2.4) into D ( ϕ ) = K ( ϕ )6 πηr k B T h (1 + 2 ϕ ) + ( ϕ − ϕ i (1 − ϕ ) . (2.6)To be in accord with Batchelor’s result (2.2) at low concentrations, the authors [3] suggested replacing ofEq.(2.2) by K ( ϕ ) ≈ (1 − ϕ ) . (2.7)which allows us to rewrite (2.6) in the following final form D/D = (1 − ϕ ) . h (1 + 2 ϕ ) + ( ϕ − ϕ i (1 − ϕ ) (2.8)convenient for comparison with experimental data.Such a comparison can be found in Fig.12 of Ref [3] where this result is plotted against author’s ownexperimental data for the cooperative diffusion coefficient. The experimental data within error margins4ppears to agree extremely well with the theoretical curve obtainable from Eq.(2.8). However, it shouldbe kept in mind that, in fact, originally Eq.(2.2) was determined only to first order in ϕ (and, therefore,only for the volume fractions less than about 0.05). Therefore, formally, Eq.(2.8) is in accord with Eq.(2.2)only for volume fractions of lesser than about 0.03. Therefore, it is clear from Fig 12 of [3] that to improvethe agreement in the whole range of concentrations, a knowledge of a second order in ϕ is desirable in(2.2). This problem can be by passed as follows.From [3 ] the viscosity data from the same experiments were obtained so that the data can be fit tothe following second order expansion: η/η = 1 + 2 . ϕ + 6 . ϕ + O (cid:0) ϕ (cid:1) . (2.9)To obtain this result, the authors constrained the first order coefficient to 2.5 to comply with Einstein’sresult (1.2) for viscosity. If one considers these data without such a constraint, then one obtains, η/η = 1 + 2 . ϕ + 7 . ϕ + O (cid:0) ϕ (cid:1) . (2.10)In the paper by Kholodenko and Douglas [23] the following result for the cooperative diffusion coeffi-cient was derived (the generalized Stokes-Einstein relation) D/D = 1( η/η ) (cid:20) S ( , S ( , (cid:21) − / , (2.11)where S ( ,
0) is the k = 0 , zero angle static scattering form factor. The thermodynamic sum rule for thehard sphere gas produces the following result for this formfactor: (cid:20) S ( , S ( , (cid:21) − / = 1 + 4 ϕ + 7 ϕ + O (cid:0) ϕ (cid:1) . (2.12)By combining Eq.s (2.9)-(2.12) the result for cooperative diffusion is obtained: D/D = 1 + 1 . ϕ − . ϕ + O (cid:0) ϕ (cid:1) . (2.13)For the sake of comparison with experiment, we made a numerical fit to the experimental data for higherconcentrations obtained in [3] by a polynomial (up to a second order in ϕ ) with the result : D/D = 1 + 1 . ϕ − . ϕ + O (cid:0) ϕ (cid:1) . (2.14)Comparison between Eq.s (2.13) and (2.14) shows that the theoretically obtained result, Eq.(2.13), is ingood agreement with the experimental data, Eq.(2.14), within error margins. Alternatively, we can usethe reciprocal of the empirical expression, Eq.(2.14), in (2.11) to obtain η/η = 1 + 2 . ϕ + 8 . ϕ + O (cid:0) ϕ (cid:1) , (2.15)which also compares well with the experimental data, Eq.(2.10). We return now to Eq.(2.11) for further discussion. Based on the results of introductory section, especiallyon Eq.s (1.1) and (1.3), we can formally write:
D/D = RR ∗ η η = 1 + a ϕ + a ϕ + O (cid:0) ϕ (cid:1) . (2.17) The correlation coefficient obtained for this fit is 0.97. a i , i = 1 , , .. can be determined using Eq.(2.11) written in the following form D/D = ( ξ/ξ ) − ( η/η ) = 1( η/η ) (cid:20) S ( , S ( , (cid:21) − / , (2.18)where ξ is the correlation length, ξ is the correlation length in the infinite dilution limit ξ ∼ R . Tojustify such a move we need to remind to our readers of some facts from the dynamical theory of linearresponse. To do so, we borrow some results from our previous work.[23].Generally, both D and D are measured by light scattering experiments. In these experiments theFourier transform of the density-density correlator S ( R , τ ) = h δn ( r , t ) δn ( r ′ , t ′ ) i (2.19)is being measured. The formfactor, Eq.(2.19), is written with account of translational invariance, re-quiring the above correlator to be a function of relative distance | r − r ′ | ≡ | R | only. Time homogeneity,makes it in addition to be a function of | t − t ′ | = τ only. In this expression, h ... i represents an equilibriumthermal average while density fluctuations are given by δn ( r , t ) = n ( r , t ) − h n i . By definition, the Fouriertransform of Eq.(2.19) is given by S ( q , ω ) = Z d r Z dτ S ( R , τ ) e i ( q · r − ωτ ) . (2.20). Using this expression, we obtain the initial decay rate Γ (0) q as follows [10,23]:Γ (0) q = − ∂∂τ ln (cid:20)Z ∞−∞ dω π e iωτ S ( q , ω ) (cid:21) τ → + . (2.21)With help of this result, the cooperative diffusion coefficient is obtained as D = ∂∂q Γ (0) q | q =0 . (2.22)In the limit of vanishingly low concentrations the self -diffusion coefficient is known to be [10] D = 16 lim t →∞ t D { r ( t ) − r (0) } E , (2.23)where < ... > denotes the Gaussian-type average. Following Lovesey [28], it is convenient to rewrite thisresult as D = 13 Z ∞ dt h v · v ( t ) i (2.24)in view of the fact that if r ( t ) − r (0) = t Z v ( τ ) dτ (2.25a)then, D ( r ( t ) − r (0)) E = * t Z v ( τ ) dτ + = 2 t Z dτ ¯ τ Z d ¯ τ h v ( τ ) · v (¯ τ ) i = 2 t Z dτ ( t − τ ) h v ( τ ) · v (0) i , (2.25b)while, by definition, D = lim t →∞ t D ( r ( t ) − r (0)) E . With these definitions in place and taking into account Eq.(2.18), we would like now to discuss inmore detail the relationship between D and D . Using Eq.(2.29) of Ref. [23] we obtain D = lim τ → + , k =0 ∞ Z dt ′′ < j ( , t ) · j ( , t ′′ ) >S ( , , (2.26)6here the current j is given as j = δn ( r , t ) v ( r, t ) , provided that the non-slip boundary condition v f ( r , t ) = d r dt ≡ v ( t ) (2.27)is applied. Here v f ( r , t ) is the velocity of the fluid and r ( t ) is the position of the center of mass of thehard sphere with respect to the chosen frame of reference. Eq.(2.26) is in agreement with Eq.(2.24) inview of the fact that in the limit of zero concentration S ( ,
0) = 1 so that < j ( , t ) · j ( , t ′′ ) > → h v · v ( t ) i as we would like to demonstrate now. For this purpose, in view of Eq.(2.22) it is convenient to rewritethe result Eq.(2.26) is the equivalent form S ( q , (0) q = ∞ Z dt ′′ q · < j ( q , t ) j ( − q , t ′′ ) > · q | τ → + (2.28)in accord with Eq.(2.15) of Ref.[23]. This relation is very convenient for theoretical analysis. For instance,it is straightforward to obtain D in the decoupling approximation as suggested by Ferrell [21]. It is givenby q · < j ( q , t ) j ( − q , t ′ ) > · q ˙= q · < v ( q ′ , t ) v ( q − q ′ , t ′ ) > · q h δn ( q − q ′ , t ) δn ( q ′ , t ′ ) i . (2.29)In Section 5.4. we provide proof that the above decoupling is in fact exact. This provides an explanationwhy it is working so well in real experiments.In the meantime, we shall consider this decoupling as an approximation. Once such an approximationis made, the problem of calculation of D is reduced to the evaluation of correlators defined in Eq.(2.29). Forthe velocity-velocity correlator, the following expression was obtained before (e.g. see Ref.[23], Eq.(2.18)): < v i k ( t ) v j k ′ ( t ′ ) > = (2 π ) δ ( k + k ′ ) { δ ij − k i k j k } k B Tηk δ ( t − t ′ ) ≡ k B T H ij ( k ) δ ( t − t ′ ) (2.30)with i, j = 1 − . This expression defines the Oseen tensor H ij ( k ) to be discussed in detail in the nextsection. The presence of the delta function δ ( t − t ′ ) in Eq.(2.30) makes it possible to look only at theequal time density-density correlator in the decomposition of the j − j correlator given by Eq.(2.29). Sucha correlator also was discussed in Ref.[23] where it is shown to be h δn ( k − k ′ , t ) δn ( k ′ , t ) i = k B T < n > (cid:20) ∂∂ Π < n > (cid:21) T (2.31a)Actually, it is both time and k − independent since, as is well known, it is the thermodynamic sum rule.That is S ( ,
0) = k B T < n > (cid:20) ∂∂ Π < n > (cid:21) T . (2.31b)It is convenient to rewrite this result as follows k B T < n > (cid:20) ∂∂ Π < n > (cid:21) T = Z d R S ( R , ≡ S ( , , (2.31c)implying that S ( R ,
0) = k B T < n > (cid:20) ∂∂ Π < n > (cid:21) T δ ( R ) . (2.32)In view of Eq.(2.12), we notice that in the limit of vanishing concentrations S ( ,
0) = 1 . In such an extremecase the decoupling made in Eq.(2.29) superimposed with the definition, Eq.(2.22), and the fact that ∂∂q · ·· = 13 tr ( X i,j ∂∂q i ∂∂q j · ·· )7roduces the anticipated result, Eq.(2.24), as required. Next, following Ferrell [21], we regularize thedelta function in Eq.(2.32). Using an identity 1 = ∞ R dxxe − x , the regularized expression for S ( R ,
0) isobtained as follows S ( r,
0) = k B T πξ < n > (cid:20) ∂∂ Π < n > (cid:21) T r e − rξ , (2.33)where r = | R | and the parameter ξ is proportional to the static correlation length . To use this expressionfor calculations of D, by employing Eq.(2.26) we have to transform the hydrodynamic correlator, Eq.(2.30),into coordinate form as well. Such a form is given in Eq.(2.33) of Ref.[23] as < v ( r , t ) · v ( r ′ , t ′ ) > = k B Tπη | r − r ′ | δ ( t − t ′ ) . (2.34)This expression is written with total disregard of possible effects of the hydrodynamic screening, though.The combined use of Eq.s (2.33) and (2.34) in Eq.(2.26) produces the anticipated result D = k B T πηξ (2.35)in accord with that obtained by Ferrell, Ref.[21], Eq.(11), provided that we redefine (still arbitrary) theparameter ξ as 2ˇ ξ . The result (2.35) also coincides with Eq.(1.3) if we identify ˇ ξ with R ∗ . Furthermore,by looking at Eq.(2.18) we realize that in the infinite dilution limit we have to replace ˇ ξ by ξ and,accordingly, η by η . Such an identification leads to the generalized Stokes-Einstein relation in the formgiven by Eq.(2.18) implying that ˇ ξ/ξ = (cid:20) S ( , S ( , (cid:21) / . (2.36a)Since we have noticed before that S ( ,
0) = 1 this result can be rewritten asˇ ξ = p S ( , ξ . (2.36b)Suppose now that hydrodynamic interactions are screened to some extent. In such a case the resultEq.(2.34) should be modified accordingly. Thus, we obtain < v ( r , t ) · v ( r ′ , t ′ ) > = k B Tπη exp( − rξ H ) r δ ( t − t ′ ) , (2.37)where we have introduced the hydrodynamic correlation length ξ H . If, as we shall demonstrate below, theanalogy between hydrodynamic and superconductivity makes sense under some conditions then, usingthis assumed analogy we introduce the Ginzburg parameter κ G for this problem via known relation [25]: ξ H = κ G ˇ ξ. (2.38)Using Eq.s (2.33), (2.37) and (2.38) in (2.26), the result for D , Eq.(2.35), acquires the following form: D = k B T π Σ η (1 + 1 κ G ) − . (2.39)Since, the adjustable parameter Σ is introduced in Eq.(2.39) quite arbitrarily, we can, following Ferrell,Ref. [21], take full advantage of this fact now. To do so, we notice that from the point of view of theobserver, the relation (2.36) holds irrespective of the absence or presence of hydrodynamic screening.Because of this, we write Σ(1 + 1 κ G ) = ˇ ξ (2.40) For more details, see our work, Ref.[23].
8o that the Eq.(2.36) used in the generalized Stokes-Einstein relation remains unchanged. By combiningEq.s(2.36b), (2.38) and (2.40) we obtain: κ G = ξ H ξ p S ( ,
0) = Σ ξ κ G p S ( ,
0) (1 + 1 κ G ) (2.41a)or, equivalently, ξ Σ = 1 p S ( ,
0) (1 + 1 κ G ) . (2.41b)In this equation the parameter Σ is still undefined. We can define this parameter now based on physicalarguments. In particular, let us set Σ = S ( , ξ . Then, we end up with the equation1 + 1 κ G = 1 p S ( ,
0) (2.42)leading to κ G = 1 √ S ( , − . (2.43)To reveal the physical meaning of this equation we use Eq.s (2.36b), (2.38) and (2.43) in order to obtain ξ H = p S ( , ξ p S ( , − . (2.44)From Eq.(2.12) we notice that by considering the infinite dilution limit ϕ → , we obtain: ξ H → ∞ , implying absence of hydrodynamic screening. Consider the opposite case: ϕ → ∞ (that is, in practice, ϕ being large). Looking at Eq.(2.12) we notice that in this case ξ H → D/D = 1( η/η ) (1 + 1 κ G ) . (2.45) If n is the local density, then the flux j = n v obeys Fick’s first law: j = − D ∇ n, (3.1)where D is the (in general, cooperative) diffusion coefficient, and v is the local velocity. Upon substitutionof this expression into the continuity equation ∂n∂t + ∇ · j = 0 (3.2)we obtain the diffusion equation commonly known as Fick’s second law ∂n∂t = D ∇ n . (3.3)9n the presence of some external forces, i.e. F = − ∇ U, (3.4)the diffusion laws must be modified. This is achieved by assuming the existence of some kind of friction,i.e.by assuming that there exists a relation γ v = F (3.5)between the local velocity v and force F with the coefficient of proportionality γ being, for instance (inthe case of hard spheres), of the type given in Eq.(1.1). With such an assumption, the diffusion current , Eq.(3.1) , is modified now as follows j = − D ∇ n − nγ ∇ U. (3.6)Such a definition makes sense. Indeed, in the case of equilibrium , when the concentration n eq obeys theBoltzmann’s law n eq = n exp( − Uk B T ) , (3.7)vanishing of the current in Eq.(3.6) is assured by substitution of Eq.(3.7) into Eq.(3.6) thus leading tothe already cited Einstein result, Eq.(1.1), for D . As in the case of Eq.(1.3), we shall assume that forfinite concentrations one can still use the Einstein-like result for the diffusion coefficient. With such anassumption, the current j in Eq.(3.6) acquires the following form [12]: j = − nγ ∇ ( k B T ln n + U ) ≡ − nγ ∇ µ, (3.8)where the last equality defines the nonequilibrium chemical potential µ, e.g. like that given in Eq.(2.1).Alternatively, the modified flux velocity v f is given now by − γ ∇ µ so that the continuity Eq.(3.2) readsas ∂n∂t + ∇ · ( n v f ) = 0 . (3.9)Exactly the same equation can be written for the probability density Ψ if we formally replace n by Ψin the above equation [10]. Such an interpretation of diffusion is convenient since it allows one to talkabout diffusion in terms of the trajectories of Brownian motion of individual particles whose positions x n ( t ) , n = 1 , , ... are considered to be as random variables. Then, the probability Ψ describes suchcollective Brownian motion process described by the following Schrodinger-like equation ∂∂t Ψ = − X n ∂∂ x n (Ψ v fn ) (3.10)in which the velocity v fn is given by v fn = − X m L nm ∂∂ x m ( k B T ln Ψ + U ) . (3.11)Thus, we obtain our final result ∂∂t Ψ = X m,n ∂∂ x n L nm ( k B T ∂∂ x m Ψ + ∂U∂ x m Ψ) (3.12)adaptable for hydrodynamic extension. For this purpose, we need to remind our readers of some basicfacts from hydrodynamics 10 .2 Hydrodynamic fluctuations and Oseen tensor
The analog of Newton’s equation for fluids is the Navier-Stokes equation. It is given by [29] ∂∂t v + ( v · ∇ ) v = − ρ ∇ P + Γ ∇ v (3.13)where P is the hydrodynamic pressure, Γ = η /ρ is the kinematic viscosity and ρ is the density of thethe pure solvent. At low Reynold’s numbers, the convective term ( v · ∇ ) v can be neglected [29], p.63. Weshall also assume that the fluid is incompressible, i.ediv v = 0 . (3.14)Under such conditions the Fourier transformed Navier-Stokes equation can be written as ∂∂t v k = − Γ k v k − i k ρ P k . (3.15)Let us add a fluctuating source term f k to the right hand side of Eq.(3.15). Then, using the incompress-ibility condition, Eq.(3.14), we obtain: P k = − iρ k · f k ( t ) k . (3.16)Introducing the transverse tensor T ij ( k ) = δ ij − k i k j k and decomposing a random force as f Ti k ( t ) = X j T ij ( k ) f j k ( t ) (3.17)eventually replaces the Navier-Stokes equation by the Langevin-type equation for the transverse velocityfluctuations: ∂∂t v k + Γk v k = f T k ( t ) . (3.18)As is usually done for such type of equations, we shall assume that the random fluctuating forces areGaussianly distributed. This assumption is equivalent to the statement that D f Ti k ( t ) f Tj k ′ ( t ′ ) E = T ij ( k )(2 π ) δ ( k + k ′ ) ˜ Dδ ( t − t ′ ) (3.19)with parameter ˜ D to be determined. A formal solution of the Langevin-type Eq.(3.18) is given by v k ( t ) = v k (0) e − ˜Γ t + t Z dt ′ f T k ( t ′ ) e − ˜Γ( t − t ′ ) (3.20)with ˜Γ = k Γ . Introducing v k ( t ) − v k (0) e − ˜Γ t = ˆv k ( t ) , we obtain h ˆ v i k ( t )ˆ v j k ′ ( t ′ ) i = * t Z dt ′ f T k ( t ′ ) e − ˜Γ( t − t ′ ) t ′ Z dt ′′ f T k ′ ( t ′′ ) e − ˜Γ( t ′ − t ′′ ) + . (3.21)To calculate this correlator, and to determine the parameter 2 ˜ D, we consider the equal time correlatorfirst. In such a case the equipartition theorem produces the following result: h ˆ v i k ( t )ˆ v j k ′ ( t ) i = T ij ( k )(2 π ) δ ( k + k ′ ) k B Tρ . (3.22)11aking into account Eq.s (3.19) and (3.22) we obtain for the velocity-velocity correlator, Eq.(3.21), thefollowing result h ˆ v i k ( t )ˆ v j k ′ ( t ′ ) i = T ij ( k )(2 π ) δ ( k + k ′ ) 2 k B Tρ ˜Γ ˜Γ2 exp( − ˜Γ | t − t ′ | )= 2 k B T H ij ( k ) ˜Γ2 exp( − ˜Γ | t − t ′ | ) . (3.23)In the limit ˜Γ → ∞ the combination ˜Γ2 exp( − ˜Γ | t − t ′ | ) can be replaced by δ ( t − t ′ ) . In this limit theobtained expression coincides with already cited Eq.(2.30). Furthermore, the constant ˜ D can be chosenas k B Tρ . To prove the correctness of these assumptions, we take a Fourier transform (in time variable) inorder to obtain (cid:10) ˆ v i k ( ω ) ˆ v j ( − k ) ( − ω ) (cid:11) ˙= 2 k B Tρ ˜Γ ω + ˜Γ T ij ( k ) . (3.24)This result coincides with Eq.(89.17) of Ref.[25] as required. Here the sign ˙= means ”up to a delta functionprefactor”. Incidentally, these prefactors were preserved in another volume of Landau and Lifshitz, e.g.see Ref. [30], Eq.(122.12). Since in the limit ω → ˜Γ2 exp( − ˜Γ | t − t ′ | ) by δ ( t − t ′ ) . In polymer physics, Ref.[10], typically only this ω → τ r = ρR /πη mentioned in the Introduction. Although this fact could cause some inconsistencies (e.g. see discussionbelow), we shall follow the traditional pathway by considering mainly this limit causing us to dropaltogether time-dependence in Eq.(3.15) thus bringing it to the form considered in the book by Doi andEdwards, Ref.[10], Eq. (3.III.2). Following these authors, this approximation allows us to specify arandom force f ( r ) as f ( r ) = X n F n δ ( r − R n ) (3.25)implying that particle (hard sphere) locations are at the points R n so that the fluctuating component offluid velocity v ( r ) at r is given by v ( r ) = X n H ( r − R n ) · F n (3.26)with the Oseen tensor H ij ( r ) in the coordinate representation given by H ij ( r ) = 18 πη | r | ( δ ij + ˆ r i ˆ r j ) . (3.27)In this expression ˆ r i = r i | r | . In view of Eq.(2.27), we can rewrite Eq.(3.26) in the following suggestive form v ( R n ) = X m ( m = n ) H ( R n − R m ) · F m , (3.28)for velocity v ( R n ) of the particle located at R n . By comparing Eq.s(3.12) and (3.28) we could write the Fick’s first law explicitly should the Oseen tensorbe also defined for m = n . But it is not defined in this case. As in electrostatics, self-interactions mustbe excluded from consideration. In view of the results of Section 2, the situation can be repaired if we12ssume that at some concentrations the hydrodynamic interactions are totally screened. In such a caseonly the usual Brownian motion of individual particles is expected to survive. With these remarks, Fick’sfirst law for such hydrodynamically interacting suspensions of spheres can be written now as follows v f ( R n ) = − X m ˜H ( R n − R m ) · ∂∂ R m ( k B T ln Ψ + U ) , (3.29)where the redefined Oseen tensor ˜H ij ( R ) has the diagonal part ˜H ii ( R ) = γ in accord with Eq.(1.1). Thepotential U comes from short-range non- hydrodynamic interactions between particles, which are alwayspresent. Using this result and Eq.(3.10), we finally arrive at the Fick’s second law ∂ Ψ ∂t = X n,m ∂∂ R n · ˜H ( R n − R m ) · ( k B T ∂ Ψ ∂ R m + ∂U∂ R m Ψ) (3.30)in accord with Eq.(3.110) of Ref.[10]. Since this equation contains both diagonal and nondiagonal termsthe question arises about its mathematical meaning. That is, we should inquire: under what conditionsdoes the solution to this equation exist? The solution will exist if and only if the above equation can bebrought to the diagonal form. To do so, as it is usually done in mathematics, we have to find generalizedcoordinates in which the above equation will acquire the diagonal form. Although the attempts to do sowere made by several authors, most notably, by Kirkwood, e.g. see Ref.[10], chr-3 and references therein,in this work we would like to extend their results to account for effects of gauge invariance.We begin with the following auxiliary problem: since ∇ =div · ∇ ≡ ∇ · ∇ , we are interested in findinghow this result changes if we transform it from the flat Euclidean space to the space described in termsof generalized coordinates. This task is easy if we take into account that in the Euclidean space ∇ · ∇ = X i,j ∂∂x i h ij ∂∂x j , (3.31)with h ij being a diagonal matrix with unit entries. We notice that the above expression is a scalar and,hence, it is covariant. This means, that we can replace the usual derivatives by covariant derivatives, themetric tensor h ij by the metric tensor g ij in the curved space so that in this, the most general case, weobtain D i g ij D j f ( x ) = ∂∂x i g ij ∂∂x j f ( x ) + g kj Γ iik ∂∂x j f ( x ) , (3.32)where summation over repeated indices is assumed, as usual. The covariant derivative D i is defined fora scalar f as D i f = ∂∂x i f and for contravariant vector X i as D j X i = ∂X i ∂x j + Γ ijk X i (3.33)with Christoffel symbol Γ ijk defined in a usual way of Riemannian geometry. A precise definition of thissymbol is going to be given below. Since Γ iik = ∂∂x k ln √ g , we can rewrite Eq.(3.32) in the followingalternative final form ∇ f = D i g ij D j f ( x ) = 1 √ g ∂∂x i [ g ij √ g ∂∂x i f ] (3.34)so that in Eq.(3.3) the operator ∇ is replaced now by that given by Eq.(3.34). To make this presentationcomplete, we have to include the relation g ij = ∂r k ∂q i ∂r l ∂q j h kl . (3.35)In the simplest case, when we are dealing with 3 dimensional vectors, so that r = r ( q , q , q ) , sometimesit is convenient to introduce vectors e i = ∂ r ∂q i (3.36)13nd the metric tensor g ij = e i · e j (3.37)with ” · ” being the usual Euclidean scalar product sign. Definitions Eq.(3.35) and (3.37) are obviouslyequivalent in the present case. Because of this, it is clear that upon transformation to the curvilinearcoordinates the Riemann curvature tensor written in terms of g ij is still zero since it is obviously zero forthe h kl . The curvature tensor will be introduced and discussed below. Before doing so, using the examplewe have just described, we need to rewrite Eq.(3.30) in terms of generalized coordinates. In the presentcase, we must have 3 N generalized coordinates and the tensor h kl is not a unit tensor anymore. Ourarguments will not change if we replace Eq.(3.30) by that in which the potential U = 0. Furthermore,we shall adsorb the factor k B T into the tensor ˜H and this redefined tensor we shall use instead of h kl . Evidently, the final result for the Laplacian, Eq.(3.34), will remain unchanged. The question arises: if inthe first example the Riemannian curvature tensor remains flat after a coordinate transformation (sincethe tensor h kl is the tensor describing the flat Euclidean space), what can be said about the Riemanntensor in the present case? To answer this question consider once again Eq.(3.35), this time with thetensor H mn instead of h ij . For the sake of argument, let us ignore for a moment the fact that each ofgeneralized coordinates is 3-dimensional. Then, we obtain, g αβ = ∂R k ∂q α ˜H kl ∂R l ∂q β , (3.38a)where we introduced a set of new generalized coordinates { Q } = { q , ..., q N } so that R l = R l ( { Q } ). Weshall use Greek indices for new coordinates and Latin for old. In the case of 3 dimensions the above resultbecomes g αβ = ∂ R k ∂ q α · ˜H kl · ∂ R l ∂ q β (3.38b)with ” · ” being the Euclidean scalar product sign as before. The indices k , l, α and β now have 3components each. We are interested in generalized coordinates which make the metric tensor g αβ diagonal.By analogy with Eq.(3.36), we introduce now a scalar product < R · R > ≡ R k · ˜H kl · R l (3.39)so that instead of the vectors e i we obtain now e α = ∂ R ∂ q α (3.40)and, accordingly, g αβ = < e α · e β > . (3.41)The Christoffel symbol can be defined now as ∂ e α ∂ q β = Γ γαβ e γ . (3.42)To find the needed generalized coordinates, we impose an additional constraint ∂ e α ∂ q β = ∂ e β ∂ q α (3.43)compatible with the symmetry of the tensor ˜H kl . By combining Eq.s(3.42) and (3.43) we obtain,Γ γαβ e γ = Γ γβα e γ (3.44)implying that Γ γαβ = Γ γβα . That is, the imposition of the constraint, Eq.(3.43), is equivalent to requiringthat our new generalized space is Riemannian (that is, without torsion). In such a space we would liketo consider the following combination R αβ ≡ ∂ e α ∂ q α ∂ q β − ∂ e β ∂ q β ∂ q α . (3.45)14gain, using Eq.(3.42) we obtain ∂∂ q α (Γ γαβ e γ ) = (cid:18) ∂∂ q α Γ γαβ (cid:19) e γ + (cid:18) ∂∂ q α e γ (cid:19) Γ γαβ . (3.46)Analogously, we obtain ∂∂ q β (Γ γβα e γ ) = (cid:18) ∂∂ q β Γ γβα (cid:19) e γ + (cid:18) ∂∂ q β e γ (cid:19) Γ γβα . (3.47)Finally, we use Eq.(3.42) in Eq.(3.46) and (3.47) in order to obtain the following result for R αβ R αβ = (cid:18) ∂∂ q α Γ γαβ (cid:19) e γ − (cid:18) ∂∂ q β Γ γβα (cid:19) e γ + Γ ωαβ Γ γωα e γ − Γ γωβ Γ ωβα e γ ≡ R γ ααβ e γ (3.48)The second line defines the Riemann curvature tensor. In the most general case it is given by R γαδβ . Bycombining Eq.s(3.40), (3.43), (3.45) and (3.48) we conclude that ∂ e α ∂ q α ∂ q β = ∂ e β ∂ q β ∂ q α (3.49)implying that the Riemann tensor is zero so that the connection Γ γαβ is flat. For such a case we canreplace the covariant derivative D i by ∇ i + A i [31]. The vector field A i is defined as follows. Introduce a1-form A via A = A i dx i , A i = A αi T α , where in the non -Abelian case T α is one of infinitesimal generatorsof some Lie group G obeying the commutation relations [ T α , T β ] = if αβγ T γ of the associated with itLie algebra. In addition, tr[ T α T β ] = δ αβ . The Chern-Simons (C-S) functional CS ( A ) producing uponminimization the needed flat connections is given by[31,32] CS ( A ) = k π Z M tr ( A ∧ d A + 23 A ∧ A ∧ A )= k π Z M ε ijk tr ( A i ( ∂ j A k − ∂ k A j ) + 23 A i [ A j , A k ]) (3.50)with k being some integer. Minimization of this functional produces an equation for the flat connections.Indeed, we have 8 πk CS ( A + B ) = Z M tr ( B ∧ d A + A ∧ B + 2 B ∧ A ∧ A )= 2 Z M tr ( B ∧ ( d A + A ∧ A ) , (3.51)where we took into account that Z M tr ( A i dx i ∧ ∂B k ∂dx j dx j ∧ dx k ) = Z M tr ( B k dx k ∧ ∂A i ∂dx j dx j ∧ dx i ) . From here, by requiring δδB CS ( A + B ) = 0 (3.52)we obtain our final result: d A + A ∧ A ≡ ( ∂A i ∂x j − ∂A j ∂x i + [ A i , A j ]) dx i ∧ dx j ≡ F ( A ) dx i ∧ dx j = 0 . (3.53)In the last equality we have taken into account that both in the C-S and Yang-Mills theory F ( A ) isthe curvature associated with connection A . Vanishing of curvature produces Eq.(3.53) for the field A .Irrespective to the explicit form of the field A , we have just demonstrated that, at least in the case whenthe potential U in Eq.(3.30) is zero, this equation can be brought into diagonal form provided that theoperator ∇ i is replaced by ∇ i + A i with the field A i to be specified below, in the next section.15 An interplay between topology and randomness: connectionswith the vortex model of superfluid He The C-S functional, Eq.(3.50), whose minimization produces Eq.(3.53) for the field A was introducedinto physics by Witten [32] and was discussed in the context of polymer physics in our previous workssummarized in Ref.[33]. Since polymer physics of fully flexible polymer chains involves diffusion-typeequations [10], the connections between polymer and colloidal physics are apparent. For this reason, wefollow Ref.[32] in our exposition and use it as general source of information.Specifically, as explained by Witten [32], theories based on the C-S functional are known as topologicalfield theories. The averages in these theories produce all kinds of topological invariants (depending uponthe generators T α in the non Abelian case) which are observables for such theories. In the present casethe question arises: should we use the non Abelian version of the C-S field theory or is it sufficient touse only its Abelian version, to be defined shortly? Since both versions of C-S theory were discussed inthe context of polymer physics in Ref.[33], we would like to argue that, for the purposes of this work, theAbelian version of the C-S theory is sufficient. We shall provide the proof of this fact in this section.The action functional for the abelian C-S field theory is given by S AC − S [ A ] = k π Z M d xε µνρ A µ ∂ ν A ρ (4.1)With such defined functional one calculates the (topological) averages with help of the C-S probabilitymeasure < · · · > C − S ≡ ˆ N Z D [ A ] exp { iS AC − S [ A ] } · · · . (4.2)The random objects which are subject to averaging are the Abelian Wilson loops W ( C ) defined by W (C) = exp { ie I C d r · A } , (4.3)where C is some closed contour in 3 dimensional space (normally, without self-intersections), and e is someconstant (”charge”) whose exact value is of no interest to us at this moment. The averages of productsof Wilson’s loops (perhaps, forming a link L ) W ( L ) = n Y i =1 W (C i ) (4.4)are the main objects of study in such a topological field theory. Substitution of W ( L ) into Eq.(4.2)produces the following result [32] < W ( L ) > C − S = exp { i (cid:18) πk (cid:19) X i,j e i e j lk ( i, j ) } (4.5)with the (Gauss) linking number lk ( i, j ) defined as lk ( i, j ) = 14 π I C i I C j [ d r i × d r j ] · ( r i − r j ) | r i − r j | ≡ π T i Z j Z ds i ds j [ v ( s i ) × v ( s j )] · ( r ( s i ) − r ( s j )) | r ( s i ) − r ( s j ) | . (4.6) E.g. see Eq.(4.12) of Ref.[33]. i and T j are respectively the contour lengths of contours C i and C j and v ( s ) = dds r ( s ) . Withthe Gauss linking number defined in such a way, in view of Eq.(4.5), it should be clear that we must toconsider as well self-linking numbers lk ( i, i ) . Such a technicality requires us to think about the so calledframing operation discussed in some detail in both Ref.s [32] and [33]. We shall ignore this technicalityuntil Section 6 for reasons which will become apparent.
Following Tanaka and Ferrari, Refs [34, 35], we rewrite the Gauss linking number in a more physicallysuggestive form. For this purpose, we introduce the ”magnetic” field B ( r ) via B ( r ) = 14 π I C j d r j × ( r − r ( s j )) | r − r ( s j ) | (4.7)allowing us to rewrite the linking number lk ( i, j ) as lk ( i, j ) = I Ci d r i · B ( r i ) . (4.8)Eq.(4.7) for the field B ( r ) is known in magnetostatics as Biot-Savart law, e.g. see Ref.[36], Eq.(30.14).Because of this, we recognize that ∇ · B =0 (4.9)and ∇ × B = j , (4.10)where j ( r ) = H C ds v ( s ) δ ( r − r ( s )) . To connect these results with hydrodynamics, we introduce the vectorpotential A in such a way that ∇ × A = B . (4.11)Using this result in Eq.(4.10) we obtain as well ∇ A = − j (4.12)in view of the fact that ∇ · A =0 . In hydrodynamics we can represent the local fluid velocity followingRef.[37], page 86, as v = ∇ × A (4.13)and define the vorticity ˜ ω as ˜ ω = ∇ × v . (4.14)By analagy with Eq.s(4.10) and (4.12) we now obtain ∇ A = − ˜ ω . (4.15)Hence, to apply previous results to hydrodynamics the following identification should be made: ˜ ω ⇄ j . ; v ⇄ B . (4.16)The kinetic energy E of a fluid in a volume M is given by E = ρ Z M v d r . (4.17)We would like now to explain how this energy is related to the above defined linking numbers. For thispurpose, we introduce the following auxiliary functional: F [ A ] i = I C i d r i · A ( r i ) . (4.18)17se of the theorem by Stokes produces F [ A ] i = I C i d r i · A ( r i ) = Z Z S i d S i · ( ∇ × A ) = Z Z S i d S i · v = Z Z S i d S i I C j ds j v ( s j ) δ ( r i − r ( s j )) . (4.19)At the same time, for the linking number, Eq.(4.8), an analogous procedure leads to the following chainof equalities lk ( i, j ) = I Ci d r i · B ( r j ) = I Ci d r i · v ( r j ) = Z Z S i d S i · ( ∇ × v ) = Z Z S i d S i · ˜ ω . (4.20a)Since the same vector potential was used in both Eq.s(4.11) and (4.13) we notice that Eq.s (4.12) and(4.15) also imply that ˜ ω = e j , (4.20b)where e is some constant to be determined. Because of this, we obtain e F [ A ] i = lk ( i, j ) = e Z Z S i d S i I C j ds j v ( s j ) δ ( r i − r ( s j )) . (4.21)Since the obtained equivalence is of central importance for the entire work, we would like to discuss a fewadditional details of immediate relevance. In particular, from Eq.(4.20b), which we shall call from nowon, the London equation (e.g. see the Subsection 4.4 below), it should be clear that the as yet unknownconstant e must have dimensionality of inverse length L − . This fact should be taken into account whenwe consider the following dimensionless functional W [ A ] = ρ k B T Z M d r ( ∇ × A ) + i ef X j I Cj d r j · A ( r j ) (4.22)and the path integral associated with it, i.e.ˇ N Z D [ A ] δ ( ∇ · A ) exp {−W [ A ] } ≡ < W ( L ) > T (4.23a)to be compared with Eq.s(4.2) and (4.5). Here the thermal average < · · · > T is defined by < · · · > T = ˇ N Z D [ A ] δ ( ∇ · A ) exp {− ρ k B T Z M d r ( ∇ × A ) } · · · . (4.23b)Calculation of this Gaussian path integral is complicated by the presence of a delta constraint (Coulombgauge) in the path integral measure. Fortunately, this path integral can be found in the paper byBrereton and Shah [38]. Without providing the details, these authors presented the following final resultin notations adapted to this work: < W ( L ) > T = exp {− ρ (cid:18) ef (cid:19) X ′ i,j =1 t Z t Z ds i ds j ˙r ( s i ) · ˜H [ r ( s i ) − r ( s j )] · ˙r ( s j ) } . (4.24)The Oseen tensor ˜H ( R ) in this expression was previously defined in Eq.(3.27) and the prime on thesummation sign means that the diagonal part of this tensor should be excluded. Even though calculationsleading to this result are not given in Ref.[38], they can be easily understood field-theoretically. For thispurpose, we have to regularize the delta function constraint in the path integral measure in Eq.(4.23) very In view of the fact that the dimensionality of e is fixed we have introduced a factor f which makes the functional W [ A ]dimensionless. This factor will be determined shortly below. N Z D [ σ ( r )] exp( − ξ Z d r σ ( r )) Z D [ A ] δ ( ∇ · A − σ ( r )) exp {− ρ k B T Z M d r ( ∇ × A ) } · · · = ˇ N Z D [ A ] exp {− ρ k B T Z M d r ( ∇ × A ) − ξ Z d r ( ∇ · A ) } · · · = ˇ N Z D [ A ] exp {− ρ k B T Z M d r A µ [ − δ µν ∇ − (1 − ξ ) ∂ µ ∂ ν ] A ν } · · · . (4.25)with some adjustable regularizing parameter ˜ ξ. Also, for the quadratic form (in A) in the exponent of thelast expression we obtain Z M d r A µ [ − δ µν ∇ − (1 − ξ ) ∂ µ ∂ ν ] A ν = Z d k A µ ( k )[ δ µν k − (1 − ξ ) k µ k ν ] A ν (4.26)The inverse of the matrix [ δ µν k − (1 − ξ ) k µ k ν ] is easy to find following Ramond [39]. Indeed, we write[ δ µν k − (1 − ξ ) k µ k ν ][ X ( k ) δ νρ + Y ( k ) k ν k ρ ] = δ µρ . (4.27)From here the unknown functions X ( k ) and Y ( k ) can be determined so the inverse matrix is givenexplicitly by [ X ( k ) δ νρ + Y ( k ) k ν k ρ ] = 1 k [ δ νρ − (1 − ˜ ξ ) k ν k ρ k ] . (4.28)In the limit ˜ ξ → /η ) in the k -space representation inaccord with Ref.[10].These results explain why in the average, Eq.(4.24), there are no diagonal terms.Now we are ready to determine the constant e introduced in Eq.(4.22). The important result for < W ( L ) > T contains random velocities ˙r ( s ) and thus, seemingly, additionalaveraging is required. The task now lies in finding the explicit form of this averaging. To do so, severalsteps are required. To begin, we notice that in the absence of hydrodynamic interactions Eq.(3.30)acquires the following form ∂ Ψ ∂t = D X n ∂ ∂ R n Ψ (4.29)with diffusion coefficient D defined in Eq.(1.1). If Eq.(4.29) we treat Ψ as Green’s function (e.g. seeAppendix A for details), then it can be formally represented in the path integral form asΨ( t ; R ,..., R N ) = Z D [ { R ( τ ) } ] exp( − D t Z X N i − [ ˙r ( τ i )] dτ i ) . (4.30)19n this expression we have suppressed the explicit R -dependence of the path integral to avoid excessivenotation. Hydrodynamic interactions can now be accounted for as follows F = ˇ N Z D [ A ] exp {− ρ k B T Z M d r A µ [ − δ µν ∇ − (1 − ξ ) ∂ µ ∂ ν ] A ν }× Z D [ { r ( τ ) } ] exp( − D t Z ( X N j − [ ˙r ( τ j )] dτ j ) exp { i ef t Z X N j − [ ˙r ( τ j )] · A [ r ( τ j )] dτ j }≡ < N Y j =1 Z D [ { r ( τ j ) } ] exp( − D t Z [ ˙r ( τ j )] dτ j ) exp { i ef t Z ˙r ( τ j ) · A [ r ( τ j )] dτ j } > T . (4.31)Perturbative calculation of path integrals of the typeI[ A ; t ] = Z D [ { r ( τ j ) } ] exp( − D t Z [ ˙r ( τ j )] dτ j exp { i ef t Z [ ˙r ( τ j )] · A [ r ( τ j )] dτ j } (4.32)was considered by Feynman long ago, Ref.[40]. From this paper it follows that the most obvious way todo such a calculation is to write the usual Schr¨odinger-like equation (cid:18) ∂∂t − D ( P − ie A ) (cid:19) G ( t, r ; t ′ r ′ ) = 0, r = r ′ (4.33)and to take into account that ( P − ie A ) = P − ie A · P − ie P · A − e A ≃ P − ie A · P + O ( A ) (since P · A =0) . This result is useful to compare with Eq.(3.32) in order to recognize that the field A is indeeda connection.To use these results, we would like to rewrite Eq.(3.30) in the alternative form which (for U = 0) isgiven by ∂ Ψ ∂t = D X n ∂ ∂ R n Ψ + k B T X ′ m,n,i,j ˜H ij ( R n − R m ) ∂∂R in ∂∂R jm Ψ. (4.34)In arriving at this equation we took into account Eq.(3.14). Consider such an equation for n = 2. In thiscase, we rewrite Eq.(4.34) in the style of quantum mechanics, i.e. (cid:18) ∂∂t − H − H − V (cid:19) Ψ = 0 (4.35)in which, as in quantum mechanics, we shall treat V as a perturbation. The best way of dealing withsuch problems is to use the method of Green’s functions. For our reader’s convenience we present somefacts about this method in Appendix A. Eq.(A.10) of this Appendix provides an equation for the effectivepotential V . A similar type of equation was obtained in the book by Doi and Edwards, Ref.[10], in Section5.7.3., who used methods of the effective medium theory. Using this theory they were able to provethe existence of screening for the case of polymer solutions. We shall reach an analogous conclusionabout screening in colloidal suspension using different arguments to be discussed in the next subsection.In the meantime, we would like to provide arguments justifying our previously made approximation:( P − ie A ) ≃ P − ie A · P + O ( A ) . Using results of Appendix A, we introduce the one-particle Green’sfunction G as a solution to the equation (cid:18) ∂∂t − D ∂ ∂ R (cid:19) G ( R , t ; R , t ′ ) = δ ( R − R ′ ) δ ( t − t ′ ) . (4.36)Having in mind the determination of previously introduced factor f (in Eq.(4.22)) , it is convenient torescale the variables in this equation to convert it into dimensionless form. Evidently, the most convenient20hoice is t = τ / ( D /R ) and R = R ˜R with R is the hard sphere radius introduced in Eq.(1.1) and τ and ˜R being dimensionless time and space coordinates. Below, we shall avoid the use of tildas for ˜R andshall still write t instead of τ . The original symbols can be restored whenever they are required. Havingthis in mind, next we consider the two-particle Green’s function G . In the absence of interactions, it isjust the product of two Green’s functions of the type given by Eq.(4.36). As a result, the Dyson-typeequation for the full Green’s function for Eq.(4.34) ( n = 2 case) is given by G ( R , R , t ; R ′′ , R ′′ , t ′′ ) = G ( R , t ; R ′ , t ′ ) G ( R , t ; R ′ , t ′ )+ Z G ( R , t ; R ′ , t ′ ) G ( R , t ; R ′ , t ′ ) V ( R ′ , R ′ ) G ( R ′ , R ′ , t ′ ; R ′′ , R ′′ , t ′′ ) d R ′ d R ′ dt ′ (4.37)in which the potential V ( R ′ , R ′ ) = k B T ˜H ij ( R − R ) ∂∂R i ∂∂R i . As before, summation over repeatedindices is assumed. Using results of Appendix A and Eq.(4.37) it is possible to write now the equationfor the effective potential. In view of the results to be discussed in the next subsection, this is actuallyunnecessary. Hence, we proceed with other tasks at this point. Specifically, taking into account Eq.(3.27)in which the explicit form of the Oseen tensor is given, we conclude that the nondiagonal part of thistensor can be discarded in the Dyson Eq.(4.37). This is so because of the following obvious identity:[( r − r ) · r ] [( r − r ) · r ] + [( r − r ) · r ] [( r − r ) · r ] = 0 associated with the scalar products ofunit vectors in Eq.(3.27). Evidently, it is always possible to select a coordinate system centered, say,at r Alternatively, this result can be easily proven in k -space taking into account the incompressibilityconstraint. Furthermore, these observations cause us to write the potential V ( R , R ) in the followingdimensionful form V ( R ′ , R ′ ) = k B T πη | R − R | ∂∂ R · ∂∂ R . (4.38a)Using dimensional analysis of Eq.(4.36), this result can be easily rewritten also in dimensionless form.Explicitly, it is given by V ( R ′ , R ′ ) = k B T πηR D | R − R | ∂∂ R · ∂∂ R (4.38b)in which the scalar product can be of any sign. This fact is of importance because of the following.Using Eq.(4.31) and proceeding with calculations of the path integral following Feynman’s prescrip-tions [40], we obtain exactly the same equation as that given by Eq.(4.37). This observation allows us todetermine the constants e and f explicitly. In view of the results just obtained, the constant e can bedetermined only with accuracy up to a sign. Taking this into account, the value of e is determined as e = ± R r D ρ πη , while the constant f is given by D in view of the fact that the field A in Eq.(4.22) hasdimensionality L /t , i.e. that of the diffusion coefficient, while the dimensionality of e is fixed by theEq.(4.20b), so that the combination eds ˙r ( s ) is dimensionless.Using these results and Eq.(4.38), we can rewrite < W ( L ) > T defined by Eq.(4.24) in the followingmanifestly dimensionless physically suggestive form < W ( L ) > T = exp( − k B TD πη X ′ i,j =1 s i R s j R I I | d r ( τ i ) · d r ( τ j ) || r ( τ i ) − r ( τ j ) | ) (4.39)where we have introduced the dimensionless Ising spin-like variables s i playing the role of charges ac-counting for the sign of the product ∂∂ R · ∂∂ R . Since the whole system must be ”electrically neutral”, atthis point it is possible to develop the Debye-H¨uckel-type theory of hydrodynamic screening by analogywith that developed for Coulombic systems, e.g. see Ref.[41]. Nevertheless, below we choose another,more elegant pathway to arrive at the same conclusions. Using dimensional analysis performed for Eq.(4.36) the result, Eq.(4.38), can be easily rewritten also in dimensionlessform D t R [ ˙r ( τ j )] dτ j present in the exponent in Eq.(4.31). While the double integral, Eq.(4.39), ismanifestly reparametrization invariant, the diffusion integral is not. This means that we can alwaysreparametrize time in this diffusion integral so that the coefficient (4 D ) − can be made equal to anypreassigned nonnegative integer. This was effectively done already when we introduced the dimensionlessvariables in Eq.(4.36). Such inequivalence between these two types of integrals can be eliminated if wereplace this diffusion -type integral by that which is manifestly reparametrization- invariant. In such acase the total action is given by S = m X i I dτ i p r ( τ i ) + k B TD πη X ′ i,j =1 s i s j I I | d r ( τ i ) · d r ( τ j ) || r ( τ i ) − r ( τ j ) | ) . (4.40)It should be noted that use of a symbol H instead of R in Eq.(4.40) is a delicate matter. In [33] wedemonstrated that in the limit of long times (that is in the limit ω → . Since the result, Eq.(4.40), is manifestly reparametrization invariant,such a replacement is permissible. Additional explanations are given in Appendix B which we recommendto read only after reading of Section 5.The constant m in Eq.(4.40) will be determined in the next section. The form of the action givenby Eq.(4.40) is almost identical to that for the action for the superfluid liquid He as discussed in thebook by Kleinert [42], page 300. From the same book, it also follows that the Ginzburg-Landau theoryof superconductivity also can be recast in the same form. We said ”almost identical to” meaning that inthese two theories (of superfluidity and superconductivity) the self-interaction of vortices is also allowedso that if the above expression would represent the dual (vortex) description of colloidal suspensiondynamics (e.g. see Appendix B), then the prime in the double summation above can be removed sincethe vortices are allowed to intersect with themselves.In the direct case, when the focus of attention is on particles, removal of the prime in the doublesummation in Eq.(4.40) would imply that the Oseen tensor is defined for particles hydrodynamicallyinteracting with themselves. This assumption is not present in the original Doi-Edwards formulation,Ref.[10]. As we noticed already in Eq.(3.29), the diagonal part of the Oseen tensor is associated with self-diffusion. The question therefore arises: can this ”almost equivalence” be converted into full equivalence?The main feature of superconductors is the existence of the Meissner (for hard spheres) and dual (forvortices) Meissner effect. In the present case such an effect is equivalent to the existence of hydrodynamicscreening. Hence, to prove such an equivalence requires us to prove the existence of hydrodynamicscreening for suspensions. Evidently, we cannot immediately use Eq.(4.40) for such a proof. Therefore,in the next subsection we use London-style arguments to arrive at the desired conclusion.
We begin our proof by taking into account the non-slip boundary condition, Eq.(2.27): v ( r , t ) = d r dt = v ( t ) . (2.27)Within the approximations made, we also have to impose the incompressibility requirement ∇ · v ( r , t ) = 0 . (3.14)Because of this requirement, the current j = ρ v becomes j = n v with the density n being a constant.Since j is a vector, we can always represent it as j = α ∇ ψ (4.41) Additional mathematical results on this property are discussed in Section 6.2. ψ and some proportionality constant α. To choose such a scalar we take intoaccount that in the present case ∇ · j =0 (4.42)implying ∇ ψ = 0 . (4.43)The vector j given by Eq .( ) is not uniquely defined. It will still obey the Eq.(4.42) if we write j = α ∇ ψ ± g A (4.44)for a vector A such that ∇ · A =0 . Evidently , a vector obeying Eq.(4.13) by construction possess thisproperty. The choice of the sign ”+” or ”-” in the above equation can be determined based on thefollowing arguments. Since j = n v and since n is constant, we can replace Eq.(4.44) by v = α ∇ ψ ± g A (4.45)by suitably redefinig constants α and g . Next, we assume that v is a random variable so that on average < v > = 0 thus implying < α ∇ ψ > ± g < A > =0 . (4.46)This equation causes us to choose the sign ”-”. After this, we can write for the correlator < v · v > = α < ∇ ψ · ∇ ψ > + g < A · A > = 2 g < A · A > . (4.47)In view of our choice of A , the < A · A > correlator coincides with that given in the exponent of Eq.(4.24).Now we take into account Eq.(4.20b) where, of course, we replace j by v so that using the dictionary,Eq.(4.16), we arrive at ˜ ω = e v (London equation) (4.48)supplemented with ˜ ω = ∇ × v (Maxwell equation). (4.14)Such an identification becomes apparent because of the following arguments. Let us use Eq.(4.45) inEq.(4.48) in order to obtain ˜ ω = e ( α ∇ ψ − e A ) . (4.49)In this equation we replaced the constant g by e . Furthermore, since Eq.(4.49) formally looks like theFick’s first law, we can as well rewrite this result as ˜ ω = e ( D π ∇ ψ − e A ) . (4.50)By applying to both sides of this equation the curl operator and taking into account Eq.(4.13), we obtain ∇ × ˜ ω = − e v . (4.51)Taking into account the Maxwell’s Eq.(4.14) and using it in Eq.(4.51) we obtain as well ∇ v = e v . (4.52a)Equivalently, we obtain, ∇ A = e A . (4.51b)Using Eq.s(4.47), (4.52a) and following the same steps as in the Appendix A of our previous work, Ref.[23],we obtain h v ( r ) · v ( ) i = constr exp( − r ξ H ) , (4.53)23here ξ H = e − = (cid:18) R r D ρ πη (cid:19) − and the constant in Eq.(4.53) can be obtained from comparisonbetween this equation and Eq.(2.37). The analogous result is also obtained for the < A · A > correlator.In accord with Eq.(2.44) we obtain the result of central importance ξ H → ∞ when ρ → , imply-ing absence of screening in the infinite dilution limit. Our derivation explains the rationale behind theidentification of Eq.s (4.14) and (4.48) with the Maxwell and London equations in the theory of super-conductivity, Ref.[25], pages 174, 175. Evidently, such an identification becomes possible only in view ofthe topological nature of the London equation, Eq.(4.48), coming from identification of Eq.(4.19) with(4.20a). In the previous section we developed a theory of hydrodynamic screening following ideas of the Londonbrothers, Ref.[24]. As is well known, their seminal work found its most notable application in the theoryof ordinary superconductors [25]. At the same time, Eq.(4.40) was originally used in the theory ofsuperfluid He. In the book by Kleinert [42] it is shown that Eq.(4.40) can be rewritten in such a waythat it will acquire the same form as used in the phenomenological Ginzburg-Landau (G-L) theory ofsuperconductivity [25]. We would like to arrive at the same conclusions differently. In doing so we alsowould like to determine both the physical and mathematical meaning of the parameter m which was leftundetermined in Eq.(4.40). We shall develop our arguments mainly following the original G-L pathway.It should be said, though, that in the present case the connections with superconductivity are only inthe structure of equations to be derived. The underlying physics is similar but not identical to that forsuperconductors. Indeed, in the case of superconductors one typically is talking about the supeconducting-to-normal transition controlled by temperature. Also, one is talking about the temperature-dependent”critical” magnetic field (the upper and the lower critical magnetic fields in the case of superconductors ofthe second kind) which destroys the superconductivity. In the present case of colloidal suspensions thereis no explicit temperature dependence: the same phenomena can take place at any temperature at whichthe solvent is not frozen. If we account for short range forces, then, of course, one can study a situationin which such a colloidal suspension is undergoing a temperature-controlled phase transition. Such a caserequires a separate treatment and will not be considered in this work. In the present case the phasediagram can be qualitatively described as follows. The infinite dilution limit corresponds to the normal state. The regime of finite concentrations corresponds to a mixed state, typical for superconductors of the second kind , and the dramatic jump in viscosity discussed in the Introduction and in Section 2 correspondsto the transition to the ”fully superconducting” state. Such a difference from the usual superconductorsbrings some new physics into play which may be useful, in other disciplines, e.g. in the high energyphysics or turbulence, etc. We begin with the one of Maxwell’s equations in its conventional form, e.g. as given in Ref.[25], page 181, ∇ × B = 4 πc j . (5.1)In the G-L theory we have for the current j the following result :j = − i ˜ e ~ m ( ϕ ∗ ∇ ϕ − ϕ ∇ ϕ ∗ ) − e mc | ϕ | A . (5.2) E.g. see Section 6. truncated ) G-L functional F [ A , ϕ ] = Z d r { ( ∇ × A ) π + ~ m (cid:12)(cid:12)(cid:12)(cid:12) ( ∇ − i ˜ e ~ c A ) ϕ (cid:12)(cid:12)(cid:12)(cid:12) } (5.3)with respect to A . Substitution of the ansatz ϕ = √ n s iψ ) into Eq.(5.2) leads to the current j = ˜ e ~ m n s ( ∇ ψ − e ~ c A ) (5.4a)to be compared with our Eq.(4.50). Evidently, this result is equivalent to the postulated London equationfor superconductors ∇ × j = − en s mc B . (5.4b)At the same time, a comparison of Eq.(5.4a) with Eq.(4.50) leads to the following chain of identifications:˜ e ~ m n s ⇄ eD and ˜ e mc n s ⇄ e . Consequently, we obtain as well: ~ m ⇄ D , ˜ en s ~ → e ; ˜ e n s m → e , c ⇄ π → π , e ~ ⇄ eD ⇄ en s . Using these identifications, we can rewrite the functional F [ A , ϕ ] as follows F [ A , ϕ ] = ρ Z d r { ( ∇ × A ) + D (cid:12)(cid:12)(cid:12)(cid:12) ( ∇ − i πeD A ) ϕ (cid:12)(cid:12)(cid:12)(cid:12) } . (5.5)In the traditional setting, the superconducting density n s is determined from the full G-L functional F [ A , ϕ ] = Z d r { ( ∇ × A ) π + ~ m (cid:12)(cid:12)(cid:12)(cid:12) ( ∇ − i ˜ e ~ c A ) ϕ (cid:12)(cid:12)(cid:12)(cid:12) + a | ϕ | + b | ϕ | ] } , (5.6)e.g. by minimization with respect to ϕ ∗ . In fact, to obtain n s it is formally sufficient to treat only thecase when A = 0. Indeed, under this condition we obtain aϕ c + b | ϕ c | ϕ c = 0 , (5.7)which has a nontrivial solution only for a < . In this case we get n s = | a | b , provided that b > , as usual.If we use this result back in Eq.(5.6), that is we use ϕ c = √ n s iψ ) in Eq.(5.6) then, the polynomial(in ϕ ) part of the functional becomes a constant. This constant is divergent when the volume of thesystem goes to infinity. To prevent this from happening another constant term is typically added to thefunctional F [ A , ϕ ] so that it acquires the following canonical form F [ A , ϕ ] = Z d r { ( ∇ × A ) π + ~ m (cid:12)(cid:12)(cid:12)(cid:12) ( ∇ − i ˜ e ~ c A ) ϕ (cid:12)(cid:12)(cid:12)(cid:12) + b | ϕ | − n s ) } . (5.8)Then, when ϕ c = √ n s iψ ) , the polynomial (in ϕ ) part of the functional vanishes and, accordingly,in this limit we require Z d r (cid:12)(cid:12)(cid:12)(cid:12) ( ∇ − i ˜ e ~ c A ) ϕ c (cid:12)(cid:12)(cid:12)(cid:12) → ~ ci e ϕ c ∇ ϕ c = A (5.10a)or to ~ c e ∇ ψ = A . (5.10b) This truncation is known in literature as the ”
London limit ”. n s was left as an adjustableparameter and, hence, microscopically undefined. This is important in our case since the phenomenonof supercoductivity can be looked upon (as in thermodynamics) without any reference to spontaneoussymmetry breaking, Higgs effect, etc. At the level of G-L theory, the London equations are reproducedwith help of the truncated G-L functional. Hence, in principle, in the present case use of the truncatedfunctional, Eq.(5.5), is also sufficient. At the macroscopic mean field level the presence of polynomialterms in the full-G-L functional, Eq.s(5.6) and (5.8) seems somewhat artificial. They do not reveal theirmicroscopic origin and are introduced just to fit the data. We would like to use some known facts fromthe path integral treatments of superconductivity/superfluidity in order to reveal their physical meaning.Such information is also useful for development of the hydrodynamic theory of colloidal suspensions. In view of Eq.(4.40), we begin our discussion with the simplest case of the path integral for a single”relativistic” scalar particle.Following Polyakov, Ref.[43], the Euclidean version of propagator for such a (Klein-Gordon) particleis given by G ( x, x ′ ) = Z (cid:18) D x ( τ ) D f ( τ ) (cid:19) exp( − m Z dτ p ˙x ( τ )) , (5.11a)where in the most general case ˙x ( τ ) = g µν ( x ) dx µ dτ dx ν dτ . (5.11b)This propagator is of interest in string theory since it represents a reduced form of the propagator for thebosonic string. As in the case of a string, the action of this path integral is manifesttly reparametrization-invariant, i.e. invariant under changes of the type x ( τ ) → x ( f ( τ )) (with f ( τ ) being some nonnegativemonotonically increasing function). The path integral measure is designed to absorb this redundancy. Thefull account of this absorption is cumbersome. Because of this, instead of copying Polyakov’s treatment ofsuch a path integral, we shall adopt a simplified treatment allowing us to recover Polyakov’s final results.We begin with an obvious well-known identity (cid:18) πt (cid:19) d exp( − x t ) = x ( t )= x Z x (0)=0 D [ x ( τ )] exp {− t Z dτ (cid:18) d x dτ (cid:19) } . (5.12)This identity is used below as follows. Consider the propagator for the Klein-Gordon (K-G) field givenby G ( x ) = Z d d k (2 π ) d exp( i k · x ) k + m . (5.13)By employing the identity 1 a = ∞ Z dx exp( − ax ) (5.14)26q.(5.13) can be rewritten as follows G ( x ) = ∞ Z dt exp( − tm ) Z d d k (2 π ) d exp( i k · x − t k )= 1 E ∞ Z dt exp( −E tm ) Z d d k (2 π ) d exp( i k · x −E t k )= 1 E ∞ Z dt exp( − t E m ) (cid:18) π E t (cid:19) d exp( − x E t )= 1 E ∞ Z dt exp( − t m ) x ( t )= x Z x (0)=0 D [ x ( τ )] exp {− E t Z dτ (cid:18) d x dτ (cid:19) } (5.15)where we used the identity, Eq.(5.12), to obtain the last line and introduced an arbitrary nonnegativeparameter E for comparison with results by Polyakov. Specifically, using page 163 of the book by Polyakov(and comparing our 3rd line above with the 3rd line of his Eq.(9.63)) we can make the following identifi-cations: E ⇄ ε , m ⇄ µ. Since, according to Polyakov, µ = ε − ( m − c √ ε ) with c being some constant,we obtain: m = E m + c √ ε . That is, the physical mass m entering the K-G equation is obtained as thelimit of the expression ( ε → m = lim m → m cr ε − ( m − m cr ) . (5.16)Clearly, such an expression is nonnegative by construction. From the last line of Eq.(5.15) it follows thatthe propagator for the K-G field is just the direct Laplace transform of the nonrelativistic ”diffusion”propagator, Eq.(5.12), with the Laplace variable m playing a role of a mass for such a field. In theEuclidean version of the K-G propagator this mass cannot be negative since in such a case the identityEq.(5.14) cannot be used so that the connection between the nonrelativistic and the K-G propagatorsis lost. However, Eq.(5.2) seemingly is for the quantum current while the propagator in Eq.(5.12) isdescribing Brownian motion, not quantum diffusion. To fix the problem we have to replace time t inEq.(5.12) by it and, accordingly, to make changes in Eq.(5.15). This then converts the Laplace transforminto the Fourier transform, provided that the nonrelativistic propagator describes the retarded Green’sfunction. To use the full strength of the apparatus of quantum field theory one needs to use the causalGreen’s functions. This is required by the relativistic covariance treating space and time coordinateson the same footing. Once all of these requirements are met, it becomes possible to treat the case of anegative mass.It should be emphasized at this point that the London-style derivation given in the previous sectionformally does not require such quantum mechanical analogy. Because of this, the following problememerges: is it possible to reproduce the functional integral F defined by Eq.(4.31) using the truncatedG-L functional for superconductivity in the exponent of the associated path integral? We would like toprovide an affirmative answer to this question now.We begin with the partition function Z for the two-component scalar K-G-type fieldln Z = − ln[det( −∇ + m )] (5.17)Since ln[det( −∇ + m )] = tr (cid:2) ln( −∇ + m ) (cid:3) (5.18)and tr (cid:2) ln( −∇ + m ) (cid:3) = Z d d k (2 π ) d ln( k + m ) , (5.19)27e can use the results of our previous work, Ref.[44], for evaluation of the last expression. Thus, weobtain, tr (cid:20) ln ( −∇ + m )( −∇ ) (cid:21) = m Z dy ddy Z d d k (2 π ) d ln( k + y )= m Z dy Z d d k (2 π ) d k + y = m Z dyG ( ; y )= ∞ Z dt m Z dy exp( − ty ) x ( t )= Z x (0)= D [ x ( τ )] exp {− t Z dτ (cid:18) d x dτ (cid:19) } = ∞ Z dtt (1 − exp( − m t )) x ( t )= Z x (0)= D [ x ( τ )] exp {− t Z dτ (cid:18) d x dτ (cid:19) } . (5.20)Following the usual practice, we shall write H instead of x ( t )= R x (0)= in the path integral and consider a formal(that is diverging!) expression for the free energy F exp ( −F ) = ln Z = − ln[det( −∇ + m )] = ∞ Z dtt exp( − m t ) I D [ x ( τ )] exp {− t Z dτ (cid:18) d x dτ (cid:19) } (5.21)by keeping in mind that this result makes sense mathematically only when the same expression with m = 0 is subtracted from it as required by Eq.(5.20). Inclusion of the electromagnetic field into thisscheme can be readily accomplished now. For this purpose we replace the ∇ operator by its covariantderivative: ∇ → D ≡ ∇ − ie A (we put D = 1 in view of developments presented in Eq.(5.15)) Using D instead of ∇ in Eq.(5.20) we have to evaluate the following path integral [det( − D + m )] − = Z D [¯ ϕ, ϕ ] exp( − Z d r { ¯ ϕ ( − D + m ) ϕ } ) . (5.22)For A = 0 we did this already while for A = 0 we can treat terms containing A as perturbation. We cando the same for the path integral in Eq.(4.32). This is easy to understand if we realize that ∞ Z dt exp( − m t )I[ A ; t ] | A =0 = − ddm ∞ Z dtt exp( − m t ) I D [ x ( τ )] exp {− t Z dτ (cid:18) d x dτ (cid:19) } (5.23)Therefore, the final answer reads as followsexp ( −F ) = ln Z = − ln[det( − D + m )] = ∞ Z dtt exp( − m t ) I D [ x ( τ )] exp {− t Z dτ [ 14 (cid:18) d x dτ (cid:19) + i ef ˙x · A [ x ( τ )]] } = ∞ Z dtt exp( − m t ) I D [ x ( τ )] exp {− t Z dτ (cid:18) d x dτ (cid:19) } exp {− i ef I d r · A [ x ( τ )] } . (5.24)This result demonstrates that applying the operator ∞ R dtt exp( − m t ) to I[ A ; t ] defined in Eq.(4.32) makesit equivalent to the ”matter” part of truncated G-L functional for superconductivity as needed. This raises For m = 0 this is just part of the truncated G-L functional.
28 question about comparison of the full G-L functional with the ”diffusion” path integrals of Section 4.Evidently, this can be done only if in the original diffusion Eq.(3.30) we do not discard the potential U . Ifwe do not discard the potential and if, instead, we ignore the hydrodynamic interactions completely, wewould end up with the following path integral for interacting Brownian particles in the canonical ensembleΞ = Z N Y l =1 D [ x ( τ l )] exp {− D N X i =1 t Z dτ i (cid:18) d x dτ i (cid:19) − N X i 2) (5.40)with linking number lk (1 , 2) defined in Eq.(4.6). If the above result is correct and the constant κ can befound then, the constant k can be determined from Eq.(5.39). Hence, the task lies in demonstrating thatthe nonzero constant κ does exist. To do so we shall use the standard London analysis. Thus, we write ∇ × B = 4 πc j (Maxwell equation) (5.1)and ∇ × j = − en s mc B (London equation). (5.4b)Since, B = ∇ × A , and ∇ · B = ∇ · A =0 , we have κ B = A so that we obtain ∇ × A = κ πc j (5.41)and, from here − ∇ A = ∇ × ∇ × A = κ πc ( ∇ × j ) = − κ πen s mc B = − κ πen s mc A (5.42)31hich is the familiar screening-type equation, e.g. see Eq.(4.51b). Since, in the conventional setting thepenetration depth δ is known to be δ = (cid:18) πen s mc (cid:19) − , we can chose κ =1 implying that k = hc e . Thechoice κ =1 does not mean of course that the constant κ is dimensionless. Because of this, we obtain14 π κ I C I C d σ d σ ′ v ( σ ) · v ( σ ′ ) | r ( σ ) − r ( σ ′ ) | = lk (1 , 2) (5.43)in accord with Eq.(4.21). Next, if we take into account screening effects, the conclusions we’ve reachedwill remain the same due to reparametrization invariance of both sides of Eq.(5.43). Indeed, consider oneloop, say C , going from - ∞ to + ∞ in the z-direction. If we compactify R by adding one point at infinityso that R becomes S , then such a loop will be closed. Another loop can stay mainly in the x-y plane sothat the linking number becomes the winding number, e.g. see Ref.[46], page 134. Under these conditionsthe screening factor exp ( − rδ ) under the integral of the left hand side of Eq.(5.43) is unimportant sincewe can always arrange our windings in such a way that r ≪ δ for any preassigned nonzero δ so that thescreening factor becomes unimportant.The above analysis can be extended to the case of colloidal suspensions in view of the results of Sections4.2 and 4.4. implying that in both superconductivity and colloidal suspensions the phase transition istopological in nature (e.g. in the colloidal case Eq.(4.39) is a topological invariant to be considered inthe next subsection). Evidently, such a conclusion cannot be reached by perturbatively calculating theGreen’s function in Eq.(4.37).In Section 5.1 we discussed similarities and differences between superconductors and colloidal sus-pensions. It is appropriate now to add a few additional details to the emerging picture. In the case ofsuperconductivity correctness of the topological picture depends upon the existence of nontrivial solu-tions of Eq.(5.42). These are possible only when the parameter n s is nonzero. When it becomes zero theabove picture breaks down. In the case of suspensions the role of the parameter κ − is played by thedensity-dependent parameter e . This can be easily seen if we take into account that dimensional analysisrequires us to replace Eq.(5.38) by A ( r ) = D π I C | r − r ( σ ) | v ( σ ) dσ (5.44)so that by employing Eq.(5.37b) we obtain, e π I C I C d σ d σ ′ v ( σ ) · v ( σ ′ ) | r ( σ ) − r ( σ ′ ) | = n = lk (1 , 2) (5.45)as expected. In the Introduction we noted that Chorin, Ref.[22], conjectured that the superfluid-to normal transitionin He is associated with vortices causing a sharp increase in viscosity. In this subsection we wold liketo demonstrate that, at least for colloidal suspensions, his conjecture is correct: the sharp increase inviscosity is associated with the lambda-type transition. Instead of treating this problem in full generality,i.e. for the nonideal Bose gas, we simplify matters and consider a Bose condensation type transitiontypical for the ideal Bose gas. It should be noted though that our simplified treatment is motivated onlyby the fact that it happens to be sufficient for comparison with experimental data. In other cases, sucha restriction can be lifted. Emergence of such a screening factor can be easily understood if we replace Eq.(4.15) by Eq.(5.42) with the right handside given by ˜ ω ( r ) = k H C v ( σ ) δ ( r − r ( σ )) in accord with Eq.(5.38). 32o develop such a theory we use the information obtained in the previous subsection augmented bysome additional facts needed for completion of our task. In particular, we are interested in the expressionfor the kinetic energy. Up to a constant it is given by E ˙= 12 Z d r ( ∇ × A ) (5.46)and is manifestly nonnegative. Using known facts from vector analysis this expression can be rewrittenas follows E ˙= 12 Z d r ( ∇ × A ) · v = 12 Z d r [ A · ˜ ω + div [ A × v ]] = 12 Z d rA · ˜ ω (5.47)In view of Eq.(5.38) we can rewrite this result as E ˙= k I C I C d σ d σ ′ v ( σ ) · v ( σ ′ ) | r ( σ ) − r ( σ ′ ) | (5.48)to be compared with Eq.(5.45). Using such a comparison we arrive at an apparent contradiction: whilean expression for E should be nonnegative, the linking number lk (1 , 2) can be both positive or negative.If we make the replacement r → − r in Eq.(5.48) nothing changes but if we do the same for lk (1 , 2) itchanges the sign. Thus, if we want to use lk (1 , 2) in Eq.(5.48) we have to use | lk (1 , | . This numberwas introduced by Arnold and is known in literature as entanglement complexity . Evidently, in viewof this remark, n in Eq.(5.45) can be only nonnegative. If we require our system to be invariant withrespect to rotations of the coordinate frame, Eq.(4.39) should be rewritten according to the proceduredeveloped in our work, Ref.[49]. This means, that we introduce a set of linking numbers: n , n , ..., n i , ... so that for a given n , the set of n ( n − ≡ N possible linking numbers can be characterized by thetotal linking number L , i.e. we have N X i =1 n i = L. (5.49)This result can be rewritten alternatively as follows. Let C be the number of links with linking number1, C the number of links with linking number 2 and so on. Then, we obtain L X i =1 iC i = L. (5.50)Furthermore, we also must require L X i =1 C i = N (5.51)Define the Stirling-type number ˜ S ( L, N ) via the following generating function L X N =0 ˜ S ( L, N ) x N = x ( x + 1) · · · ( x + L − . (5.52)Set in this definition x = 1. This then allows us to introduce the probability p ( L, N ) = ˜ S ( L, N ) /L ! Thenumber ˜ S ( L, N ) can be easily obtained with the result given by˜ S ( L, N ) = N Y i =1 L ! i C i C i ! . (5.53) For more deatails about this number and its many applications can be found in our works, Refs.[47,48]. E.g. see Eq.(4.4). The true Stirling number of the first kind S ( L, N ) is defined as follows: S ( L, N ) := ( − L − N ˜ S ( L, N ) . E.g. see Ref. [49]. r ( τ ) → R ˜r ( τ ), with ˜r ( τ ) being dimensionless. After which, using Eq.(5.45) we can rewrite Eq.(4.39) asfollows < W ( L ) > T = exp( − η η L ) (5.54)Evidently, the numerical factor of 3 in the exponent is non-essential and can be safely dropped uponrescaling of L . To use this expression we combine it with Eq.(5.34) in which we have to make someadjustments following Feynman, Ref.[50], pages 62-64. On these pages Feynman discusses a partitionfunction for the ideal Bose gas written in the path integral form. We would like to rewrite his result inthe notation of our paper. For this purpose we use Eq.(4.30) in which the path integral is written for aloop and is in discrete form. We obtain, h ( ν ) = (cid:18) πD (cid:19) ν Z ν Y i =1 d r i exp {− D [( r − r ) + ( r − r ) + ... + ( r ν − − r ν ) + ( r ν − r ) } = V (cid:18) πνD (cid:19) ν V = R d r . Under such circumstances the Brownian ring is made out of ν links(segments) so that wecan identify its length with ν. In the present case each such ring is linked with another ring thus forminga link with a linking number iC i , i = 0 , , , ... Since the linking number is independent of the lengthsof rings from which it is made, we can take advantage of this fact by identifying the index i with ν. Bycombining Eq.s (5.50)-(5.55) and repeating the same steps as given in Feynman’s lectures we assemblethe following dimensionless grand partition function F e −F = X C ,...,C q ,.... Y ν h ( ν ) C ν C ν ! ν C ν exp( − η η νC ν ) = X C ,...,C q ,.... Y ν C ν ! ( h ( ν ) z ν ν ) C ν = Y ν ∞ X C ν =0 C ν ! ( h ( ν ) z ν ν ) C ν = exp( X ν h ( ν ) z ν ν ) . (5.56)Here the ”chemical” potential z = exp ( − η η ) . Taking the logarithm of both sides of the above equationwe obtain the partition function for the ideal Bose gas. Written per unit volume it reads F = − (cid:18) πD (cid:19) ζ / ( z ) . (5.57)In this expression ζ α ( z ) is Riemann’s zeta function ζ α ( z ) = ∞ X n =1 z n n α . (5.58)This function is well defined for z < 1, i.e. for ηη < ∞ and is divergent for z > , thus indicating a Bosecondensation whose onset is determined by the value z = 1 (i.e. η = ∞ ) for which ζ / (1) = 1 . . If wefollow standard treatments, then we obtain for the critical density ρ c ρ c = (cid:18) πD (cid:19) 32 2 . . (5.59)In view of Eq.(5.55), the obtained result for density has the correct dimensionality. From here the criticalvolume fraction is: ϕ c = ρ c πR . The number 2 . 612 is just the value of ζ / (1) . This means, that we can34rite in the general case ρ ( z ) = (cid:18) πD (cid:19) ζ / ( z ) (5.60)thus giving us the equation ρ c − ρρ c = 1 − ζ / ( z ) ζ / (1) . (5.61)In the book by London, Ref.[51], we found the following expansion for ζ / ( z ) in the vicinity of z = 1( z < 1) : ζ / ( z ) = − . α + 2 . 612 + 1 . α − . α + ...., (5.62)where α = − ln z. Use of this result in Eq.(5.61) produces the following result: ηη = (cid:18) . (cid:19) (1 − ρρ c ) − (5.63)in accord with scaling predictions by Brady, Ref. [19], and Bicerano et al. Ref.[20]. It should be notedthough that in view of Eq.(5.54) the actual value of the constant prefactor in Eq.(5.63) is quite arbitraryand can be adjusted with help of experimental data. For instance, by making this prefactor of orderunity, Bicerano et all obtained a very good agreement with experimental data in the whole range ofconcentrations, e.g. see Ref. [20], Fig.4. With the exception of the work by De Gennes [52] on phase transition in smectics A, the superconductiv-ity and superfluidity phenomena are typically associated with the domain of low temperature condensedmatter physics .This fact remains true even with account of cuprate superconductors, Ref.[54]. Theresults obtained in this work cause us to look at these phenomena differently. For instance, the previouslymentioned relation ˜ ω ( r ) = k H C v ( σ ) δ ( r − r ( σ )) used in the work by Lund and Regge, Ref.[45], for fluids,coincides with our Eq.(4.20b) for colloids. The work of Lund and Regge is based on previous work byRasetti and Regge, Ref.[55], on superfluid He and, therefore, their results are apparently valid only inthe domain of low temperatures.This conclusion is incorrect however as shown in the series of papersby Berdichevsky, Ref.s [56,57]. Any ideal (that is Euler-type) incompressible fluid can be treated thisway. Furthermore, as results by Chorin, Ref.[22], indicate, the same methods should be applicable fordescription of the onset of fluid/gas turbulence. In our work the fluid is manifestly nonideal. Nevertheless,in the long time (zero frequencies) limit it can still be treated as if it is ideal.The most spectacular departure from traditional view on the results by Lund and Regge was recentlymade in a series of papers by Schief and collaborators, Refs.[58,59]. The latest results elaborating onhis work can be found in Ref.[60]. Schief demonstrated that the results of Lund and Regge work well inthe case of magnetohydrodynamics, that is, ultimately in the plasma installations designed for controlledthermonuclear synthesis.The basic underlying physics of all these phenomena can be summarized as follows. In every systemwhich supports knotted structures, the existence of a decoupling of topological properties from the confor-mational (statistical) properties of flux tubes from which these knots/links are made should be possible.Since this statement is not restricted to a simple Abelian C-S field theory describing knots/links existingin G-L theory, in full generality the theory should include the G-L theory as a special case (as demon-strated above). Accordingly, the minimization of the corresponding truncated G-L functional may ormay not lead to London-type equations. We would like to illustrate these general statements by specificexamples. This is accomplished below. Lately, however, these ideas have began to be popular in color supercoductivity dealing with quark matter [53]. .2 Helicity and force-free fields imply knoting and linking but not nesess-sarily superconductivity via London mechanism The concept of helicity has its origin in theory of neutrino, Ref.[61]. An expression σ · p / | p | is called helicity. Here σ · p = σ x p x + σ y p y + σ z p z , and p i and σ i , i = 1 − , are being respectively the componentsof the momentum and Pauli matrices. The eigenvalue equation[ σ · p / | p | ] Ψ = λ Ψ (6.1)produces eigenvalues λ which can be only ± . Moffat, Ref.[62], designed a classical analog of the helicityoperator. He proposed to use the product v · ∇ × v ≡ v · ˜ ω for this classical analog. In it, as before, e.g.see Eq.(4.14), the vorticity field ˜ ω is used. Moffat constructed an integral (over the volume M ) I = Z M v · ˜ ω dV (6.2)along with two other integrals: the kinematic kinetic energy2 Tρ E = Z M v dV (6.3)and the rotational kinetic energy Ω = Z M ˜ ω dV. (6.4)Then, he used the Schwarz inequality I ≤ E Ω or Ω ≥ I E (6.5)in order to demonstrate that the equality is achieved only if ˜ ω = α v where α is a constant. Since thisrequirement coincides exactly with our Eq.(4.20b), it is of interest to sudy this condition further. Inparticular, under this condition we obtain αI = E which would coincide with our Eq.(5.43) (see also5.48)) should I be associated with the linking number. Fortunately, this is indeed the case. The proofwas given by Arnold and is outlined in Ref.[63], pages 141-146. In view of its physical significance, wewould like to discuss it in some detail.Before doing so, we notice that the condition ˜ ω = α v is known in literature as the force-free condition for the following reason. In electrodynamics, the motion of an electron in a magnetic field is given by (inthe system of units in which m = c = e = 1) d v dt = v × B (6.6a)while the use of the Maxwell’s equation, our Eq.(4.10), produces as well v = ∇ × B = α B (6.6b)Using previously established equivalence v ⇄ B and substitution of Eq.(6.6b) into Eq.(6.6a) explainswhy the force-free condition is given by ˜ ω = α v . This equation can be looked upon as an eigenvalueequation for the operator ∇ × ( · · · ) . From this point of view the force-free equation is totally analogousto its quantum counterpart, Eq.(6.1). Details can be found in Ref.[64].Going back to Arnold’s proof, we note that according to Moffatt, Ref.[62], page 119, I = Z V v · ˜ ω dV = 14 π Z V (1) Z V (2) R · [˜ ω (1) × ˜ ω (2)] k R k dV (1) dV (2) . (6.7)36learly, if as is done by Moffatt and others in physics literature (e.g. Lund and Regge, etc.), we assumethat the vector potential A can be given in the form of Eq.(5.44), then I indeed becomes the linkingnumber, Eq.(4.6). If, however, we do not make such an assumption, then much more sophisticatedmethods are required for the proof of this result. Use of these methods is not of academic interest only,as we would like to explain now. According to Kozlov, Ref.[65],the force-free case ˜ ω = α v belongs to thecategory of so called vortex motion in the weak sense. There are many other vortex motions for which v ×∇× v = . These are vortex motions in the strong sense . Evidently, any relation with superconductivityor superfluidity (which is actually only hinted at this stage in view of results obtained in previous sections)is lost in this (strong) case. But even with the vorticity present in the weak sense this connection is notimmediately clear.This is so because of multitude of solutions of the force-free equation as discussed, forexample, in Refs.[66, 67]. We would like to discuss only those solutions which are suitable for use inArnold’s theorem. These solutions can be obtained as follows. Taking the curl of the equation ∇ × B = α B , (6.8)provided that ∇ · B = 0 , produces ( ∇ + α ) B = 0 , (6.9)to be compared with our result, Eq.(4.52a). Unlike our case, which is motivated by analogies with super-conductivity and superfluidity, in the present case there are many solutions of this equation. We chooseonly the solution which illustrates the theorem by Arnold. It is given by v = ( Asinz + Ccosy, Bsinx + Acosz, Csiny + Bcosx ) , where ABC = 0 and A, B, C ∈ R This solution is obtained for α = 1 . Following Arnold, we introduce the asymptotic linking number Λ( x , x ) viaΛ( x , x ) = lim T ,T →∞ πT T T Z T Z dt dt ( ˙x ( t ) × ˙x ( t )) · ( x ( t ) − x ( t )) k x ( t ) − x ( t ) k . (6.10a)The theorem proven by Arnold states that if the motion described by trajectories x ( t ) and x ( t ) isergodic, then14 π Z V (1) Z V (2) R · [˜ ω (1) × ˜ ω (2)] k R k dV (1) dV (2) = 1 V Z V (1) Z V (2) Λ( x , x ) dV (1) dV (2) = lk (1 , . (6.10b)That is the function Λ( x , x ) on ergodic trajectories is almost everywhere constant. This theorem assuch does not imply that this constant is an integer. For us it is important to realize that both Eq.(4.52a)and Eq.(6.9) can produce trajectories minimizing the Schwarz inequality thus leading to the condition αI = E with I being either linking (in the case of suspensions) or self-linking number (depending uponthe problem in question) or a conbination of both. Because both Eq.(4.52a) and (6.9) cause formationof links, the choice between them should be made on a case-by-case basis. In particular, existence of theMessner effect in superconductors leaves us with no freedom of choice between these two equations. Inthe case of magnetohydrodynamics/plasma physics the situation is less obvious. In the next subsectionwe shall argue in favour of superconducting/superfluid choice between these equations. To our knowledge,such a choice was left unused in plasma physics literature. In order to discuss the work by Schief, Ref.[58], we would like to remind to our readers of some factsfrom the work by Lund and Regge (originally meant to describe superfluid He) since these fact nicelysupplement those presented in previous sections. We already mentioned that Berdichevsky adopted theseresults for normal fluids, including those which are turbulent. Lund and Regge assumed that the vortexhas a finite thickness so that the non-slip boundary condition, Eq.(2.27), should be now amended toaccount for finite thickness. The amended equation is given by v i ( t ) = ∂x i ∂t + ∂x i ∂σ ∂σ∂t , (6.11)37here σ parametrizes the coordinate along the vortex line. Eq.(5.38) taken from work by Lund and Reggethen implies: ε ijk ∂x j ∂σ ( ∂x k ∂t − v k ) = 0 . (6.12)This equation is treated as an equation of motion by Lund and Regge obtained with help of the followingLagrangian L = kρ Z C ε ijk x i ∂x j ∂σ ∂x k ∂t dσ − ρ Z V v d V. (6.13)Since the zero thickness limit of the action for this Lagrangian is given by our Eq.(4.22), which uponintegration of the A-field leads to the result, Eq.(4.24), the same can be done in the present case and,accordingly, by analogy with the action, Eq.(4.22), which was extended, e.g. see Eq.(4.40), in the presentcase it can be extended as well so that the final result for the action of the Nambu-Goto bosonic stringinteracting with electromagnetic-type field reads (using the same signature of space-time as used inRef.[45]) S = − m Z dσdτ √− g + f Z A µν ∂x µ ∂σ ∂x ν ∂t dσdτ − Z F dvol (6.14)with √− g = [ − (cid:18) ∂x ν ∂σ ∂x ν ∂σ (cid:19) · (cid:18) ∂x µ ∂τ ∂x µ ∂τ (cid:19) + (cid:18) ∂x ν ∂σ ∂x ν ∂τ (cid:19) ] . (6.15)and m and f being some coupling constants. The metric of the surface enclosing the vortex can bealways brought to diagonal form by some conformal transformation . In such coordinates, variation ofthe action S produces the following set of equations m ( ∂ ∂τ − ∂ ∂σ ) x µ = f ε µνλρ F ν ∂x ρ ∂τ ∂x λ ∂σ (6.16a)and ∂ µ ∂ µ A αβ = − f Z dσdτ ( ∂x α ∂σ ∂x β ∂τ − ∂x α ∂τ ∂x β ∂σ ) δ (4) ( x ( σ, τ ) − y ) (6.16b)provided that ∂ µ A µν = 0 . Since the last equation is just the wave equation with an external source, theequation of motion for the vortex is Eq.(6.16a). In such a form it was obtained in Ref.[58] describingvortices in ideal magnetohydrodynamics. Under some physically plausible condition it was reduced in thesame reference to the equation of motion for the one-dimensional Heisenberg ferromagnet. This resultwill be discussed further below from a somewhat different perspective.It should be noted though that Eq.(6.16a) emerges in Ref.[58] under somewhat broader conditionsthan those allowed by the force-free equation. In view of the content of the next subsection, we wouldlike to reproduce this, more general case, now. For this purpose, we recall that the Euler’s equation forfluids can be written in the form, Ref.[29], ∂∂t ˜ ω = ∇ × ( v × ˜ ω ) . (6.17)In the case when ˜ ω is time-independent, it is sufficient to require only that v × ˜ ω = ∇ Φ (6.18)with Φ being some (potential) scalar function. In the case of hydrodynamics the equation Φ = const isthe famous Bernoulli equation. Thus, the force-free condition in this case is equivalent to the Bernoullicondition/equation. In magnetohydrodynamics there is an analog of the Bernoulli equation as explainedin Ref.[69]. So, again, the equation Φ = const is equivalent to the force-free condition. In the case ofmagnetohydrodynamics the vortex Eq.(6.16.a) is obtained under the condition Φ = const. Since Eq.(6.16a)describes the vortex filament, the helicity integral, Eq.(6.7), describes either linking, self-linking or both. For more details, please see Ref.[68]. 38n the case of self-linking it is known, e.g. see Ref.s[48,63], that lk (1 , 1) = T w + W r. Analytically, thewrithe W r term is expressible as in Eq.(4.6) but with C and C now representing the same closed curve.The need for T w disappears if the closed curve can be considered to have zero thickness. More accurately,the closed curve should be a ribbon in order to have a nonzero T w . This is explained in Ref.[63]. Withthe exception of Appendix C, in this work we have ignored such complications. Euler’s Eq.(6.17) can be rewritten in the equivalent form: ∂∂t v = v × ˜ ω − ∇ Φ . (6.19)Following Kozlov, Ref.[65], in the case of Hamiltonian mechanics it is convenient to consider a very similar(Lamb) equation given by ∂∂t u + ( ∇ × u ) · v = − ∇ Φ , (6.20)in which the vector u is such that ∇ · u = 0 . It can be demonstrated that Hamiltonian dynamics isisomorphic to the dynamics described by the above Lamb equation, provided that we make the followingidentifications. Let Σ nt be a manifold in phase space P = T ∗ M admitting a single-valued projection ontoa configurational space M . In canonical coordinates x and y this manifold is defined by the equation y = u ( x , t ) . (6.21)It is not difficult to demonstrate that the manifold Σ nt is an invariant manifold for a canonical Hamiltonian H ( x , y , t ) if and only if the field y satisfies the Lamb’s Eq.(6.20) and that Φ( x , t ) = H ( x , y ( x , t ) , t ) is afunction on M parametrized by time t in such a way that v = ∂H∂ y | y = u (6.22)and ˙y = − ∂H∂ x | y = u = ∂ u ∂t + ∂ u ∂ x · v . (6.23)Relevance of these results to our discussion can be seen when Eq.(6.23) is compared with Eq.(6.11) ofLund and Regge. This comparison shows their near equivalence. In view of this, we would like to exploitthis equivalence further by employing it for analysis of the truncated G-L functional analogous to ourEq.(5.3) typically used for phenomenological description of high temperature superconductors [54]. Inthis case the functional F [ A , ϕ ] should be replaced by˜ F [ A , ϕ ] = Z d r { ( ∇ × A ) π + ~ m ⊥ (cid:12)(cid:12)(cid:12)(cid:12) ( ∇ ⊥ − i ˜ e ~ c A ⊥ ) ϕ (cid:12)(cid:12)(cid:12)(cid:12) + ~ m k (cid:12)(cid:12)(cid:12)(cid:12) ( ∇ k − i ˜ e ~ c A k ) ϕ (cid:12)(cid:12)(cid:12)(cid:12) } (6.24)with its components lying in the x-y (cuprate) plane and z-plane perpendicular to it. By varying thisfunctional with respect to A ⊥ and A k separately we obtain respectively the following components for theMaxwell’s equation ∇ × B i = 4 πc j i ( i = ⊥ and k ) , (6.25)where j ⊥ = − ie ~ m ⊥ ( ϕ ∗ ∇ ⊥ ϕ − ϕ ∇ ⊥ ϕ ∗ ) − e m ⊥ c | ϕ | A ⊥ . and j k = − ie ~ m k ( ϕ ∗ ddz ϕ − ϕ ddz ϕ ∗ ) − e m k c | ϕ | A k . (6.26)39rom here we obtain the phenomenological London-type equations ∇ × j ⊥ = − en s m ⊥ c B ⊥ and ∇ × j k = − en s m k c B k . (6.27)By combining Eq.s (6.25) and (6.27) and using results of our Sections 4.2. and 4.4 we can rewrite theseequations in the following suggestive (London-type) form ˜ ω ⊥ = e ⊥ v ⊥ and ˜ ω k = e k v k . (6.28)This form allows us to make a connection with the inertial dynamics of a nonrigid body. Following Kozlov,Ref.[65], we consider the motion of a nonrigid body in which particles can move relative to each other dueto internal forces. Let the inertia axes of the body be the axes of the moving frame. Let K be the angularmomentum of the body relative to a fixed point and ω the angular velocity of the moving trihedron whilethe inertia matrix I is diag ( I ⊥ , I ⊥ , I k ) . The angular momentum and the angular velocity are related by K = I ω + λ , (6.29)where λ =( λ ⊥ , λ ⊥ , λ k ) is the gyroscopic torque originating from the motion of particles inside the body.From here we obtain the Euler equation ˙K = ω × K =0 , (6.30)which is a simple consequence of Eq.(6.29). In view of Eq.s(4.45) and (4.48) we can identify Eq.s (6.28)with (6.29) thus formally making Eq.s (6.29), of London type. The hydrodynamic analogy can be in factextended so that the hydrodynamically looking Lamb-type equation can be easily obtained and analyzed.Details are given in Ref.[65], page 148. At this point our readers may have already noticed the following. 1.In our derivation of Eq.(5.63) wemade screening effects seemingly disappear while the title of our work involves screening. 2.In Eq.(6.14)we introduced the Nambu-Goto string normally used in hadron physics associated with non AbelianYang-Mills (Y-M) gauge fields. Quantum chromodynamics (QCD) of hadrons and mesons is definitelynot the same thing as scalar electrodynamics (that is G-L model) discussed in our work. 3. Variationof the action S in Eq.(6.14) leading to the string equation of motion, Eq.(6.16a), under some conditionsreduces to the equation of motion for the Heinsenberg (anti) ferromagnetic chain, which indeed describesthe motion of the vortex filaments [59]. From this reference it follows that such equation of motion, inprinciple, can be obtained quite independently from the Nambu-Goto string, QCD, etc. In this subsectionwe demonstrate that the above loose ends are in fact indicative of the very deep underlying mathematicsand physics needed for a unified description of all of these phenomena.The formalism developed thus far in this work suffers from a kind of asymmetry. On one hand, westarted with a solution of hard spheres and then we noticed that these hard speres in solution act ascurrents (if one is using the magnetic analogy). The famous Biot-Savart law in magnetostatics is causingtwo currents to be entangled with each other thus creating the Gauss linking number, Eq.(4.8). Thus, itappears that in solution two particles (currents) are always linked (entangled) with each other. That thisis indeed the case was noticed long ago as mentioned in the Introduction, e.g. see Ref.[11]. We can treatthe vortices causing such linkages as independent objects. This is reflected in the fact that we introducedthe vorticity ˜ ω ( r ) as ˜ ω = k H C dσ v ( σ ) δ ( r − r ( σ )), e.g read comments after Eq.(5.38). In view of our majorequation ˜ ω ( r ) = e v , we can think either about the velocity (or vorticity) of a particular hard sphere orabout the velocity of a particular vortex. Because of this, it is possible to treat both particles and vortices For the sake of comparison with superconductors, we deliberately choose the matrix in such form. 40n the same footing. In such a picture (sketched in Appendix B) one can either eliminate vortices andthink about effective interactions between hard spheres or vice versa. In this sense we can talk about the duality of descriptions and, hence, about the dual Meissner effect -for loops instead of particles .Before describing the emerging picture in more detail, we note the following. Consider the expressionfor vorticity ˜ ω = k H C dσ v ( σ ) δ ( r − r ( σ )) from the point of view of reparametrization invariance. Inparticular, since we have a closed contour, we can always choose it as going from infinity to minus infinity(it is easy topologically to wrap it onto a closed contour of any size). For the function y = exp ( σ ) wehave evidently 0 ≤ y ≤ ∞ when σ varies from - ∞ to ∞ . This means that we can replace σ by ln y in theexpression for the vorticity in order to obtain ˜ ω = k Z −∞ dz v ( z ) δ ( r − r ( z )) , (6.31)which in a nutshell is the same thing as a Dirac monopole, Ref.[70], with charge strength k , so that thevortices can be treated as Dirac monopoles. In Appendix C we provide some facts about Dirac monopolesin relation to vortices. According to Dirac [70] the string attached to such a monopole can either go toinfinity (as in the present case) or to another monopole of equal and opposite strength. In our case thismeans only that when two hard spheres become hydrodynamically entangled, they cannot escape thelinkage they formed. This is the (topological) essence of quark confinement in QCD known as monopolecondensation . But we are not dealing with QCD in this work! How then we can talk about the QCD?The rationale for this was put forward first by Nambu, Ref.[71] . In his work he superimposed the G-L andDirac monopole theories to demonstrate quark confinement for mesons (these are made of just two quarks:quark and antiquark). For this qualitative picture to make sense, there should be some way of reducingQCD to G-L type theory. The feasibility of such an abelian reduction (projection) was investigatedfirst by ’t Hooft in Ref.[72]. Recent numerical studies have provided unmistakable evidence supportingthe idea of quark confinement through monopole condensation, Ref.s [73,74]. Theoretical advancementsmade since the publication of ’t Hooft’s paper took place along two different (opposite) directions. In onedirection, recently, Faddeev and Niemi found knot-like topological solitons using a Skyrme-type nonlinearsigma model and conjectured that such a model can correctly represent QCD in the low energy limit[75,76]. That this is indeed the case was established in a series of papers by Cho [77,78] and, morerecently, by Kondo, Ref.[79]. In another direction, in view of the fact that, while macroscopically theMeissner effect is triggered by the effective mass of the vector field, microscopically this mass is generatedby Cooper pairs [25], it makes sense to look at detection of the excited states of such Cooper-like pairsexperimentally. The famous variational BCS treatment of superconductivity contains at its heart the gapequation responsible for the formation of Cooper pairs. The BCS treatment of superconductivity wassubstantially improved by Richardson, Ref.[80], who solved the microscopic model exactly. His modelis known in literature as the Richardson model. Closely related to this model is a model proposed byGaudin. It is also exactly solvable (by Bethe anstatz methods) [81]. The Gaudin model(s) describesvarious properties of one dimensional spin chains in the semiclassical limit. Energy spectra of the Gaudinand Richardson models are very similar. In particular, under some conditions they are equidistant,like those for bosonic string models. . Recently, we were able to find new models associated withVeneziano amplitudes, e.g. see Ref.s [82,83], describing meson-meson scattering processes. In particular,we demonstrated that the Richardson-Gaudin spin chain model producing equidistant spectra can beobtained directly from Veneziano amplitudes. Since the Veneziano amplitudes describe extremely wellthe meson mass spectrum, and since we demonstrated that the Richardson-Gaudin model (originally usedin superconductivity and nuclear physics) can be recovered from combinatorial and analytical properties It should be noted that in the case of usual superconductors one should distinguish between the constant magnetic fieldspenetrating superconductors and the fields made by vortices. In the case of colloidal suspensions it is also possible to createsome steady velocity current and to consider velocity at a given point in the fluid as made of both steady and fluctuatingparts. That is the Bose-Einstein-type condensation in view of results of Section 5. This is explained further in Appendix C We discuss his work briefly in Appendix C Also, for monopoles models discussed in Appendix C. 41f these amplitudes, this means that the Abelian reduction can be considered as confirmed (at least formesons) not only numerically but also experimentally. In Section 3.3. we demostrated that for colloidal suspensions it is sufficient to use only the Abelian versionof the Chern-Simons theory for description of emerging entanglements. There could be other instanceswhere such an Abelian treatment might fail. Examples of more sophisticated non-Abelian fluids wereconsidered in several recent excellent reviews [84,85]. These papers might serve as points of departure forthe treatment of more elaborate hydrodynamical problems involving non-Abelian entanglements. Finally,the force-free equation ˜ ω = α v which is used in our work, is known to possesss interesting new physicalproperties when, instead of treating α as a constant, one treats α as some function of the coordinates.Such treatment can be found in Ref.[86] and involves the use of conformal transformations and invariantsrecently considered in our work on the Yamabe problem, Ref.[87], and the Poincare ′ conjecture, Ref.[68]. Acknowledgement . Both authors gratefully acknowledge useful technical correspondence and con-versations with Dr. Jack Douglas (NIST). This work would look very different or even would not bewritten at all without his input and his impatience to see this work completed. Appendix A. Some facts from the theory of Green’s functions Consider an equation (cid:18) ∂∂t − H (cid:19) Φ = 0 . (A.1)Such an equation can be written in the form of an integral equation as followsΦ( x , t ) = Z G ( x , t ; x ′ , t ′ )Φ ( x ′ , t ′ ) d x ′ dt ′ (A.2)so that Φ( x , t → t ′ ) = Φ ( x , t ′ ) . (A.3)Under such conditions, the Green’s function G ( x , t ; x ′ , t ′ ) must obey the following equation (cid:18) ∂∂t − H (cid:19) G ( x , t ; x ′ , t ′ ) = δ ( x − x ′ ) δ ( t − t ′ ) (A.4)provided that G = 0 for t < t ′ . In a more complicated situation, when (cid:18) ∂∂t − H − V (cid:19) G ( x , t ; x ′ , t ′ ) = δ ( x − x ′ ) δ ( t − t ′ ) (A.5)we can write a formal solution for G in the form of the integral (Dyson’s) equation G ( x , t ; x ′ , t ′ ) = G ( x , t ; x ′ , t ′ ) + Z G ( x , t ; x ′ , t ′ ) V ( x ′ , t ′ ) G ( x ′ , t ′ ; x ′′ , t ′′ ) d x ′ dt ′ (A.6)or, symbolically, G = G + G V G . In the case of Eq.(4.35) of the main text, we have to replace Eq.(A.5)by (cid:18) ∂∂t − H − H − V (cid:19) G ( x , x , t ; x ′ , x ′ , t ′ ) = δ ( x − x ′ ) δ ( x − x ′ ) δ ( t − t ′ ) (A.7)42nd, accordingly, the Dyson type Eq.(A.6) is now replaced by the analogous equation in which nowwe must have G ( x , x , t ; x ′ , x ′ , t ′ ) = G ( x , t ; x ′ , t ′ ) G ( x , t ; x ′ , t ′ ) . To check the correctness of sucha decomposition we note that for x = x ′ Eq.s (A.1) and (A.4) coincide while for t → t ′ integration ofEq.(A.4) over a small domain around zero and taking into account that G = 0 for t < t ′ produces G ( x , t → t ′ ; x ′ , t ′ ) = δ ( x − x ′ ) . Repeating these arguments for the two-particle Green’s function andusing Eq.(A.7) (with V = 0 ) provides the needed proof of the decomposition of G in the two-particlecase.Define now formally the renormalized potential V via G = G + G V G . (A.8)Then, by comparing this equation with the original Dyson’s equation for G we obtain G − G = G V G = G V G = G V ( G + G V G ) (A.9)This allows us to write the integral equation for the effective potential V as V = V + V G V . (A.10) Appendix B. Dual treatment of the dynamics of colloidal supensions and hydrodynamicscreening We begin by first considering screening. The path integral for the functional, Eq.(5.5), can be conve-niently rewritten as follows F [ A , ϕ ] = ρ Z d r { ( ∇ × A ) + D (cid:12)(cid:12)(cid:12)(cid:12) ( ∇ − i πeD A ) ϕ (cid:12)(cid:12)(cid:12)(cid:12) } = ρ Z d r { ( ∇ × A ) + (cid:18) D π (cid:19) ( ∇ ψ − πeD A ) } (B.1)upon substitution of the ansatz ϕ = √ D π exp( iψ ) into first line of Eq.(B.1). Such a substitution isconsistent with the current defined in Eq.(4.50).Since ∇ · A =0 , we obtain( ∇ ψ − πeD A ) = ( ∇ ψ ) + (cid:18) πeD A (cid:19) − πeD A · ∇ ψ. (B.2)Consider now the following path integral Z = Z D { ψ } exp[ − (cid:18) D π (cid:19) Z d r (( ∇ ψ ) − πeD A · ∇ ψ )] . (B.3a)Since it is of a Gaussian-type, it can be straightforwardly calculated with the result Z = N exp( − e A µ ∂ µ ∂ ν ∇ A ν ) . (B3b)Here N is some (normalization) constant. Using this result and Eq.(B.1) we obtain the following finalexpression for the partition function for the vector A-field with account of constraintsΞ = Z D [ A ] exp {− ρ k B T Z d r { A µ [ − δ µν ∇ − (1 − ξ ) ∂ µ ∂ ν ] A ν + e A µ ( δ µν − ∂ µ ∂ ν ∇ ) A ν } . (B.4a)This result is in complete accord with Eq.(4.51b) where for the mass m of the vector field A we obtained: m = e . The above derivation was made without the use of Higgs-type calculations , Ref[88]. Surely, it is43n accord with these calculations. We would like now to rewrite the obtained result in a somewhat formal(simplified) form as follows:Ξ = Z D [ A ] δ ( ∇ · A ) exp {− ρ k B T Z d r [ ( ∇ × A ) + e A ] } . (B.4b)This will be used below in such simplified form. To avoid extra notation, we also set ρk B T = 1 . Thisfactor can be restored if needed.Now we are ready for the dual treatment, which can be done in several ways. For instance, followingthe logic of Dirac’s paper [70 ], we replace Ξ byΞ = Z D [ A ] δ ( ∇ · A ) exp {− Z d r [ ( ∇ × A ) + v ) + e A ] } (B.5)where v = ˜ ω e = H C dσ v ( σ ) δ ( r − r ( σ )) . Next, we use the Hubbard-Stratonovich-type identity allowing usto make a linearization, e.g.exp {− Z d r [( ( ∇ × A ) + v ) } = Z D [ Ψ ] exp[ − Z d rΨ + i Z d r ( ( ∇ × A ) + v ) · Ψ ] (B.6)Then, we take advantage of the fact that ( ∇ × A ) · Ψ = ( ∇ × Ψ ) · A + ∇ · ( A × Ψ ).By ignoring surfaceterms this allows us to rewrite the above result as follows Z D [ Ψ ] exp[ − Z d rΨ + i Z d r ( ( ∇ × A ) + v ) · Ψ ]= Z D [ Ψ ] exp[ − Z d rΨ + i Z d r ( ( ∇ × Ψ ) · A + v · Ψ )] (B.7)Using this result in Eq.(B.5) and using the Hubbard-Stratonovich transformation again we obtain:Ξ = Z D [ A ] δ ( ∇ · A ) exp {− Z d r [ ( ∇ × A ) + v ) + e A ] } = Z D [ Ψ ] δ ( ∇ · Ψ ) exp[ − Z d r [ Ψ + 1 e ( ∇ × Ψ ) ] + i Z d rΨ · v ] . (B.8)Since exp( i R d rΨ · v ) = exp( i H C dσ v ( σ ) · Ψ ( σ )) we can use this expresiion in Eq.(5.34) in order eventuallyto arrive at the functional of G-L-type (analogous to Eq.(5.6) with obviously redefined constants). Thevector field Ψ is now massive. It is convenient to make a replacement: Ψ ⇄ e Ψ to make the functionalfor the Ψ field look exactly as in Eq.(B.5) (with v = 0). The above transformations provide a manifestlydual formulation of the colloidal suspension problem. These transformations can be made differentlynevertheless. Such an alternative treatment is useful since the end result has relevance to string theoryand to the problem of quark confinement in QCD as was first noticed by Nambu, Ref.[71]. This topic isdiscussed briefly in the next appendix. Appendix C Nambu string and colloidal suspensions: Some unusual uses of Diracmonopoles. 44e begin with Eq.(B.5) but this time we treat it differently. In particular, we haveΞ = Z D [ A ] δ ( ∇ · A ) exp {− Z d r [ ( ∇ × A ) + v ) + e A ] } = Z D [ A ] δ ( ∇ · A ) exp {− Z d rv − Z d r [ ( ∇ × A ) · v − Z d r ( ∇ × A ) − e Z d rA } = Z D [ A ] δ ( ∇ · A ) exp {− Z d rv − Z d r [ ( ∇ × v ) · A − Z d r ( ∇ × A ) − e Z d rA } = Z D [ A ] δ ( ∇ · A ) exp {− Z d rv + e X i