Field-Theoretic Thermodynamic Uncertainty Relation -- General formulation exemplified with the Kardar-Parisi-Zhang equation
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Field–Theoretic Thermodynamic UncertaintyRelation
General formulation exemplified with the Kardar–Parisi–Zhang equation
Oliver Niggemann · Udo Seifert
Received: date / Accepted: date
Abstract
We introduce a field-theoretic thermodynamic uncertainty relationas an extension of the one derived so far for a Markovian dynamics on a discreteset of states and for overdamped Langevin equations. We first formulate aframework which describes quantities like current, entropy production anddiffusivity in the case of a generic field theory. We will then apply this generalsetting to the one-dimensional Kardar-Parisi-Zhang equation, a paradigmaticexample of a non-linear field-theoretic Langevin equation. In particular, wewill treat the dimensionless Kardar-Parisi-Zhang equation with an effectivecoupling parameter measuring the strength of the non-linearity. It will beshown that the field-theoretic thermodynamic uncertainty relation holds upto second order in a perturbation expansion with respect to a small effectivecoupling constant.
Keywords field theory · non-equilibrium dynamics · thermodynamicuncertainty relation · Kardar-Parisi-Zhang equation
The thermodynamic uncertainty relation (TUR) in a non-equilibrium steadystate (NESS) provides a bound on the entropy production in terms of meanand variance of an arbitrary current [1]. Specifically, in the NESS, after atime t a fluctuating integrated current X ( t ) has a mean h X ( t ) i = j t , and O. NiggemannII. Institute for Theoretical Physics, University of Stuttgart, Pfaffenwaldring 57, 70550Stuttgart, GermanyE-mail: [email protected]. SeifertII. Institute for Theoretical Physics, University of Stuttgart, Pfaffenwaldring 57, 70550Stuttgart, GermanyE-mail: [email protected] Oliver Niggemann, Udo Seifert a diffusivity D = lim t →∞ (cid:10) ( X ( t ) − j t ) (cid:11) / (2 t ). With the entropy productionrate σ the expectation of the total entropy production in the NESS is given by σ t . These quantities satisfy the universal thermodynamic uncertainty relation σ ≥ j D , i.e. σ is bounded from below by j /D . The TUR has been proven for a Marko-vian dynamics on a general network by Gingrich et al. [2,3] and further inves-tigated for a number of different settings, both in the classical (see, e.g., [4,5,6,7,8,9,10,11,12,13,14,15]) and the quantum domain (see, e.g., [16,17,18,19,20,21,22]). It has led to a deeper understanding of systems far from equi-librium as it introduces a lower bound on the dissipation given the knowledgeof the occurring fluctuations. Such a relation is of interest for the modelingand analysis of e.g. biomolecular processes, which may often be described asa Markov network (see e.g. [23,24,25]).Of particular interest is the work by Gingrich et al. [8], where the authorsextend the relation from mesoscopic Markov jump processes to overdampedLangevin equations. Here a temporal coarse-graining procedure is described,which allows the formulation of a discrete Markov jump process in terms of anoverdamped Langevin equation for the mesoscopic states of the model. Theseauthors observe that for purely dissipative dynamics the TUR is saturated.An additional spatial coarse-graining performed in [8] results in a macroscopicdescription, where it is found that the tightness of the resulting uncertaintyrelation increases with the strength of the Gaussian potential wells (see [8],fig. 9).In this work, we present a field-theoretic equivalent to the TUR. Such a ther-modynamic uncertainty relation for general field-theoretic Langevin equationsmay prove helpful in further understanding complex dynamics like turbu-lence for fluid flow or non-linear growth processes, described by the stochasticNavier-Stokes equation (e.g. [26]) or the Kardar-Parisi-Zhang equation [27],respectively. Both are prominent representatives of field-theoretic Langevinequations. For the latter, we highlight the recent progress concerning a studyof the inward growth of interfaces in liquid crystal turbulence as an experi-mental realization. On the theory side, analytic results on the effect of agingof two-time correlation functions for the interface growth were found [28]. Fur-thermore we refer the reader to three review articles [29,30,31] concerning thelatest developments around the Kardar-Parisi-Zhang universality class.The paper is organized as follows. In order to state a field-theoretic version ofthe thermodynamic uncertainty relation, we translate in section 2 the notion ofcurrent, diffusivity and entropy production known from the setting of coupledLangevin equations to their respective equivalents for general field-theoreticLangevin equations. As an illustration of the generalizations introduced insection 2, we will then study the one-dimensional Kardar-Parisi-Zhang (KPZ)equation as a paradigmatic example of such a field-theoretic Langevin equa-tion. As the calculation of the current, diffusivity and entropy production in ield–Theoretic Thermodynamic Uncertainty Relation 3 the NESS requires a solution to the KPZ equation, we will use spectral the-ory and construct an approximate solution in the weak-coupling regime of theKPZ equation in section 3. With this approximation, we will then derive insection 4 the thermodynamic uncertainty relation to quadratic order in thecoupling parameter. In this section, we will present a generalization of the thermodynamic un-certainty relation introduced in [1] to a field theory. Consider a generic fieldtheory of the form ∂ t Φ γ ( r , t ) = F γ [ { Φ µ ( r , t ) } ] + η γ ( r , t ) , h η γ ( r , t ) i = 0 , h η γ ( r , t ) η κ ( r ′ , t ′ ) i = K ( r − r ′ ) δ γ, κ δ ( t − t ′ ) . (1)Here Φ γ ( r , t ) is a scalar field or the γ -th component of a vector field ( γ ∈ [1 , n ]; n ∈ N ) with r ∈ Ω ⊂ R d , F γ [ { Φ µ ( r , t ) } ] represents a (possibly non-linear) functional of Φ µ and η γ ( r , t ) denotes Gaussian noise, which is white intime, and with K ( r − r ′ ) as spatial noise correlations. Prominent examples of(1) are the stochastic Navier-Stokes equation for turbulent flow (see e.g. [26])or the Kardar-Parisi-Zhang equation for non-linear growth processes [27] toname only two. The latter will be treated in the subsequent sections withinthe framework established in the following.Let us begin with the introduction of some notions. A natural choice of alocal fluctuating current j ( r , t ) is j ( r , t ) ≡ ∂ t Φ ( r , t ) , (2)with Φ ( r , t ) = ( Φ ( r , t ) , . . . , Φ n ( r , t )) ⊤ . The local current j ( r , t ) is fluctuatingaround its mean, i.e. j ( r , t ) = h j ( r , t ) i + δ j ( r , t ) , (3)with δ j ( r , t ) denoting the fluctuations. Given that the system (1) possesses aNESS, the long-time behavior of the local current (2) can be described as j ( r , t ) = J ( r ) + δ j ( r , t ) , (4)with J ( r ) = lim t →∞ ∂ t h Φ ( r , t ) i = lim t →∞ h Φ ( r , t ) i t , (5)where h·i denotes averages with respect to the noise history. As the thermo-dynamic uncertainty relation in a Markovian network is formulated for someform of integrated currents, we define in analogy the projection of the localcurrent onto an arbitrarily directed weight function g ( r ) j g ( t ) ≡ Z Ω d r j ( r , t ) · g ( r ) . (6) Oliver Niggemann, Udo Seifert
The integral in (6) represents the usual L -product of the two vector fields j ( r , t ) and g ( r ) with j ( r , t ) · g ( r ) = P k j k ( r , t ) g k ( r ) as the scalar productbetween j and g . With this projected current j g ( t ), we associate a fluctuating‘output’ Ψ g ( t ) ≡ Z Ω d r Φ ( r , t ) · g ( r ) . (7)Hence j g ( t ) = ∂ t Ψ g ( t ) and in the NESS J g ≡ lim t →∞ h Ψ g ( t ) i t . (8)The fluctuating output Ψ g ( t ) provides us with the means to define a measure ofthe precision of the system output, namely the squared variational coefficient ǫ , as ǫ ≡ D ( Ψ g ( t ) − h Ψ g ( t ) i ) E h Ψ g ( t ) i . (9)If the system is in its non-equilibrium steady state, we can rewrite (9) as ǫ = D ( Ψ g ( t ) − J g t ) E ( J g t ) . (10)Let us now connect the variance of the output Ψ g ( t ) to the Green-Kubo diffu-sivity given by D g ≡ Z ∞ dt h δj g ( t ) δj g (0) i . (11)Using (6) and (2), it is straightforward to verify that Z t dt ′ δj g ( t ′ ) = e Ψ g ( t ) − D e Ψ g ( t ) E , e Ψ g ( t ) ≡ Ψ g ( t ) − Ψ g (0) . Thus, (cid:28)(cid:16) e Ψ g ( t ) − D e Ψ g ( t ) E(cid:17) (cid:29) = Z t dr Z t ds h δj g ( r ) δj g ( s ) i . (12)By dividing both sides of (12) by 2 t and taking the limit of t → ∞ it is foundin analogy to [32], thatlim t →∞ R t dr R t ds h δj g ( r ) δj g ( s ) i t = D g , with D g from (11) and therefore D g = lim t →∞ (cid:28)(cid:16) e Ψ g ( t ) − D e Ψ g ( t ) E(cid:17) (cid:29) t . (13) ield–Theoretic Thermodynamic Uncertainty Relation 5 Since in the NESS Ψ g ( t ) is stochastically independent of the initial configu-ration Ψ g (0), we can simplify the expression for the diffusivity in the NESSaccording to D g = lim t →∞ D ( Ψ g ( t ) − h Ψ g ( t ) i ) E t . (14)With the result of (14) and ǫ from (9), an alternative formulation of theprecision in a NESS is ǫ = D ( Ψ g ( t ) − h Ψ g ( t ) i ) E h Ψ g ( t ) i = 2 D g J g t . (15)We proceed with expressing the total entropy production ∆s tot . The totalentropy production is given by the sum of the entropy dissipated into themedium along a single trajectory, ∆s m , and the stochastic entropy, ∆s , ofsuch a trajectory; see e.g. [33]. The medium entropy is given by, ∆s m ≡ ln p [ Φ ( r , t ) | Φ ( r , t )] p [ e Φ ( r , t ) | e Φ ( r , t )] . (16)Here p [ Φ ( r , t ) | Φ ( r , t )] denotes the functional probability density of the entirevector field Φ ( r , t ), i.e. the field configuration after some time t has elapsedsince a starting-time t < t , conditioned on an initial value Φ ( r , t ), i.e. a cer-tain field configuration at the starting time t . In contrast, p [ e Φ ( r , t ) | e Φ ( r , t )]is the conditioned probability density of the time reversed process, i.e. startingin the final configuration at time t and ending up in the original one at time t . For the sake of simplicity, we will write in the following p [ Φ ] and p [ e Φ ] in-stead of p [ Φ ( r , t ) | Φ ( r , t )] and p [ e Φ ( r , t ) | e Φ ( r , t )], respectively. The functionalprobability density can be expressed via a so-called action functional, S [ Φ ],according to p [ Φ ] ∝ exp [ −S [ Φ ]] . (17)For the system (1), the action functional (see e.g. [33,34,35,36,37,38,39,40]and references therein) is given by S [ Φ ] = 12 X γ Z tt dt ′ Z d r (cid:16) ˙ Φ γ ( r , t ′ ) − F γ [ { Φ µ ( r , t ′ ) } ] (cid:17) × Z d r ′ K − ( r − r ′ ) (cid:16) ˙ Φ γ ( r ′ , t ′ ) − F γ [ { Φ µ ( r ′ , t ′ ) } ] (cid:17) , (18)where K − ( r − r ′ ) is the inverse of the noise correlation kernel K ( r − r ′ ) from(1). The two integral kernels fulfill Z d r ′′ K ( r − r ′′ ) K − ( r ′′ − r ′ ) = δ d ( r − r ′ ) . (19)Note, that we do not explicitly state an expression for the Jacobian ensuingfrom the variable transformation η ( r , t ) → Φ ( r , t ) in (18). This is justified as Oliver Niggemann, Udo Seifert we will use the action functional to derive a general expression for the mediumentropy where it turns out that the Jacobian does not contribute (s.f. [33,41]).Inserting (17), (18) into (16) and noticing that only the time-antisymmetricpart of the action functional (18) and its time-reversed counterpart survives,leads to (see also [33,41,42]) ∆s m = 2 X γ Z tt dt ′ Z d r Z d r ′ ˙ Φ γ ( r , t ′ ) K − ( r − r ′ ) F γ [ { Φ µ ( r ′ , t ′ ) } ] . (20)The stochastic entropy change ∆s for the same trajectory, is given by (see also[41]) ∆s ≡ − ln p [ Φ ( r , τ )] (cid:12)(cid:12)(cid:12) tt . (21)Thus, the total entropy production ∆s tot reads ∆s tot = 2 X γ Z tt dt ′ Z d r Z d r ′ ˙ Φ γ ( r , t ′ ) K − ( r − r ′ ) F γ [ { Φ µ ( r ′ , t ′ ) } ] − ln p [ Φ ( r , τ )] (cid:12)(cid:12)(cid:12) tt . (22)With (22) we may also define the rate of total entropy production σ in a NESSaccording to σ = lim t →∞ h ∆s tot i t . (23)The expressions stated in (9) and (22) provide us with the necessary ingredi-ents to formulate the field-theoretic thermodynamic uncertainty relation as h ∆s tot i ǫ = 2 D g σJ g ≥ , (24)with σ from (23), D g from (13) and J g from (8). The higher the precision,i.e. the smaller ǫ , the more entropy h ∆s tot i is generated, i.e. the higher thethermodynamic cost. Or, in other words, in order to sustain a certain NESScurrent J g , a minimal entropy production rate σ ≥ J g /D g is required. Within this section we will lay the groundwork for the calculation of the quan-tities entering the TUR for the KPZ equation. The main focus thereby is onthe perturbative solution of the KPZ equation in the weak-coupling regimeand the discussion of issues with diverging terms due to a lack of regularity. ield–Theoretic Thermodynamic Uncertainty Relation 7 , b ], b > η ( x, t ) ∂ h ( x, t ) ∂t = ˆ L h ( x, t ) + λ (cid:18) ∂ h ( x, t ) ∂x (cid:19) + η ( x, t ) h η ( x, t ) i = 0 h η ( x, t ) η ( x ′ , t ′ ) i = ∆ δ ( x − x ′ ) δ ( t − t ′ ) , (25)subject to periodic boundary conditions and, for simplicity, vanishing initialcondition h ( x,
0) = 0, x ∈ [0 , b ] (i.e. the growth process starts with a flatprofile). Here ˆ L = ν∂ x is a differential diffusion operator, ∆ a constant noisestrength, and λ the coupling constant of the non-linearity.A Fourier-expansion of the height field h ( x, t ) and the stochastic driving force η ( x, t ) reads h ( x, t ) = X k ∈ Z h k ( t ) φ k ( x ) ,η ( x, t ) = X k ∈ Z η k ( t ) φ k ( x ) . (26)The set of { φ k ( x ) } is given by φ k ( x ) ≡ √ b e πikx/b k ∈ Z , (27)and thus h k ( t ), η k ( t ) ∈ C in (26). A similar proceeding for the case of theEdwards-Wilkinson equation was used in [43,44,45,46]. Inserting (26) into(25) leads to X k ∈ Z ˙ h k ( t ) φ k ( x )= X k ∈ Z h k ( t ) ˆ Lφ k ( x ) + λ X l,m ∈ Z h l ( t ) h m ( t ) ∂ x φ l ( x ) ∂ x φ m ( x ) + X k ∈ Z η k ( t ) φ k ( x )= X k ∈ Z h k ( t ) µ k φ k ( x ) − π λb X l,m ∈ Z l m h l ( t ) h m ( t ) φ l ( x ) φ m ( x )+ X k ∈ Z η k ( t ) φ k ( x ) , with { µ k } defined as µ k ≡ − π νb k k ∈ Z . (28) Oliver Niggemann, Udo Seifert
For the { φ k ( x ) } the relation φ l ( x ) φ m ( x ) = φ l + m ( x ) / √ b holds and thus thedouble-sum in the Fourier expansion of the KPZ equation can be rewritten inconvolution form setting k = l + m . This yields X k ∈ Z ˙ h k ( t ) φ k ( x )= X k ∈ Z h k ( t ) µ k φ k ( x ) − π λb / X k,l ∈ Z l ( k − l ) h l ( t ) h k − l ( t ) φ k ( x )+ X k ∈ Z η k ( t ) φ k ( x ) , (29)which implies ordinary differential equations for the Fourier-coefficients h k ( t ),˙ h k ( t ) = µ k h k ( t ) − π λb / X l ∈ Z l ( k − l ) h l ( t ) h k − l ( t ) + η k ( t ) . (30)The above ODEs (30) are readily ‘solved’ by the variation of constants formula,which leads for flat initial condition h k (0) ≡ h k ( t ) = Z t dt ′ e µ k ( t − t ′ ) η k ( t ′ ) − π λb / X l ∈ Z \{ } l ( k − l ) h l ( t ′ ) h k − l ( t ′ ) , (31) k ∈ Z . Note, that the assumption of flat initial conditions is not in conflictwith (21) as in the NESS, in which the relevant quantities will be evaluated,the probability density becomes stationary. With (31), a non-linear integralequation for the k -th Fourier coefficient has been derived. In subsection 3.4,the solution to (31) will be constructed by means of an expansion in a smallcoupling parameter λ . We close this section with the following general remarks.(i) Equation (31) has been derived on a purely formal level. In particular, theintegral R dt ′ e µ k ( t − t ′ ) η k ( t ′ ) has to be given a meaning. In a strict mathematicalformulation, this integral has to be written as Z t e µ k ( t − t ′ ) dW k ( t ′ ) , (32)which is called a stochastic convolution (see e.g. [47,48,49,50]). This has itsorigin in the fact that the noise η ( x, t ) in (25) is mathematically speaking a gen-eralized time-derivative of a Wiener process W ( x, t ) (see also subsection 3.2,(35)). In this spirit, (31) with the first integral on the right hand side replacedby (32) may be called the mild form of the KPZ equation (in its spectral rep-resentation) and h ( x, t ) = P k ∈ Z h k ( t ) φ k ( x ), h k ( t ) solution of equation (31), isthen called a mild solution of the KPZ equation. In mathematical literature,proofs of existence and uniqueness of such a mild solution can be found forvarious assumptions on the regularity of the noise (see e.g. [49,51,52] and ref-erences therein). An assumption will be adopted (see subsection 3.2), whichguarantees the existence of k h ( x, t ) k L ([0 ,b ]) , i.e. the norm on the Hilbert space ield–Theoretic Thermodynamic Uncertainty Relation 9 of square-integrable functions L . This norm, or respectively the corresponding L -product, denoted in the following by ( · , · ) , of h with any L -function g , i.e.( h, g ) , will be used in subsection 4.1 and subsection 4.3 to calculate the nec-essary contributions to a field-theoretic thermodynamic uncertainty relation.Furthermore, with this assumption on the noise, it is shown in Appendix Cfor the mild solution that almost surely h ( x, t ) ∈ C ([0 , T ] , L ([0 , b ])), T > t h ( x, t ) is a continuous function in time t with values h ( · , t ) ∈ L ([0 , b ]). This justifies the choice H = L ([0 , b ]) in the following cal-culations.(ii) The Fourier expansion applied above can be understood in a more generalsense. For the case of periodic boundary conditions, the differential operatorˆ L possesses the eigenfunctions { φ k ( x ) } and corresponding eigenvalues { µ k } from (27) and (28), respectively. It is well-known that the set { φ k ( x ) } con-stitutes a complete orthonormal system in the Hilbert space L (0 , b ) of allsquare-integrable functions on (0 , b ). Thus the Fourier-expansion performedabove can also be interpreted as an expansion in the eigenfunctions of theoperator ˆ L .(iii) With this interpretation, (31) also holds for a ‘hyperdiffusive’ version ofthe KPZ equation in which the operator ˆ L is replaced by ˆ L p ≡ ( − p +1 ∂ px ,with p ∈ N and adjusted eigenvalues { µ pk } . This may be used to introduce ahigher regularity to the KPZ equation.(iv) Besides the complex Fourier expansion in (26) with coefficients h k ( t ) ∈ C ,the real expansion h ( x, t ) = P k ∈ Z e h k ( t ) γ k ( x ), e h k ( t ) ∈ R (e.g. [49]) and γ = 1 √ b , γ k = r b sin 2 πk xb , γ − k = r b cos 2 πk xb k ∈ N , (33)will be used in the next section. The relationship between h k ( t ) and e h k ( t )reads h k ( t ) = e h − k ( t ) − i e h k ( t ) √ , h − k ( t ) = e h − k ( t ) + i e h k ( t ) √ h k ( t ) , (34)with h k ( t ) as the complex conjugate.3.2 A Closer Look at the NoiseIn the following discussion of the noise it is instructive to pretend, for thetime being, that the noise is spatially colored with noise correlator K ( x − x ′ )instead of assuming directly spatially white noise.The noise η ( x, t ) is given by a generalized time-derivative of a Wiener process W ( x, t ) ∈ R [47,48,49,26], i.e. η ( x, t ) = p ∆ ∂ W ( x, t ) ∂t . (35) Such a Wiener process W ( x, t ) can be written as (e.g. [47,49]) W ( x, t ) = X k ∈ Z α k β k ( t ) γ k ( x ) . (36)Here { α k } ∈ R are arbitrary expansion coefficients that may be used to intro-duce a spatial regularization of the Wiener process, { β k ( t ) } ∈ R are stochas-tically independent standard Brownian motions and { γ k ( x ) } from (33). Awell-known result for the two-point correlation function of two stochasticallyindependent Brownian motions β k ( t ) reads [47] h β k ( t ) β l ( t ′ ) i = δ k,l ( t ∧ t ′ ) , (37)with ( t ∧ t ′ ) = min( t, t ′ ).In the following it will be shown that the noise η defined by (35) and (36)possesses the autocorrelation h η ( x, t ) η ( x ′ , t ′ ) i = K ( x − x ′ ) δ ( t − t ′ ) , (38)which for K ( x − x ′ ) = ∆ δ ( x − x ′ ) results in the one assumed in (25). Further-more, an explicit expression of the kernel K ( x − x ′ ) by means of the Fouriercoefficients { α k } of W ( x, t ) from (36) will be given.To this end, first an expression for the two-point correlation function of theWiener process itself can be derived according to h W ( x, t ) W ( x ′ , t ′ ) i = t ∧ t ′ b " α + X k ∈ N (cid:2) α − k + α k (cid:3) cos 2 πk x − x ′ b + X k ∈ N (cid:2) α − k − α k (cid:3) cos 2 πk x + x ′ b . (39)To represent the noise structure dictated by (25), the expression in (39) hasto be an even, translationally invariant function in space. Thus, the followingrelation has to be fulfilled α − k = α k ∀ k ∈ N . (40)Then the two-point correlation function of the Wiener process is given by h W ( x, t ) W ( x ′ , t ′ ) i = t ∧ t ′ b " α + 2 X k ∈ N α k cos 2 πk x − x ′ b . (41)With W ( x, t ) = P k ∈ Z W k ( t ) φ k ( x ), φ k ( x ) from (27), equation (41) implies forthe two-point correlation function of the Fourier coefficients W k ( t ) h W k ( t ) W l ( t ′ ) i = α k α l δ k, − l ( t ∧ t ′ ) , k, l ∈ Z . (42)This result leads immediately to h η k ( t ) η l ( t ′ ) i ≡ ∆ ∂ h W k ( t ) W l ( t ′ ) i ∂t ∂t ′ = ∆ α k α l δ k, − l δ ( t − t ′ ) , k, l ∈ Z , (43) ield–Theoretic Thermodynamic Uncertainty Relation 11 using ∂ t ∂ t ′ ( t ∧ t ′ ) = δ ( t − t ′ ).For the relation between (41) and the noise from (38), we differentiate (41)with respect to t and t ′ yielding h η ( x, t ) η ( x ′ , t ′ ) i = ∆ ∂ h W ( x, t ) W ( x ′ , t ′ ) i ∂t ∂t ′ = ∆ b " α + 2 X k ∈ N α k cos 2 πk x − x ′ b δ ( t − t ′ ) . (44)The following identification can be made K ( x − x ′ ) = ∆ b " α + 2 X k ∈ N α k cos 2 πk x − x ′ b = K ( | x − x ′ | ) , (45)which structurally represents the standard implicit assumption that K ( x − x ′ )is translationally invariant, positive definite and even. Note, that the regularityof the noise-kernel K ( | x − x ′ | ) is given by the behavior of the set of { α k } for k → ∞ , where { α k } are the dimensionless Fourier coefficients of the underlyingWiener process from (36) for all k . For the case of α k = 1 ∀ k ∈ Z , spatiallywhite noise is obtained.Thus, the derivation via the Wiener process has indeed led to a translationallyinvariant real-valued two-point correlation function for η ( x, t ), given by (38),with K ( | x − x ′ | ) from (45), which describes white in time and spatially coloredGaussian noise. In the following, we will use (45) to approximate spatiallywhite noise to meet the required form in (25).Now the assumption mentioned in the remarks in subsection 3.1 can be mademore precise. In the following it will be assumed that (see Appendix C) X k ∈ Z k ǫ α k < ∞ , ǫ > . (46)This assumption excludes white noise for k ∈ N , but via the introduction of acutoff parameter Λ ∈ N , Λ ≫ k ,white noise is accessible, i.e. for k ∈ R with R ≡ [ − Λ, Λ ] . (47)Note that for the linear case, i.e. the Edwards-Wilkinson model, the authorsof [44] also introduce a cutoff, albeit in a slightly different manner. Such acutoff amounts to an orthogonal projection of the full eigenfunction expan-sion of (25) to a finite-dimensional subspace spanned by the eigenfunctions φ − Λ ( x ) , . . . , φ Λ ( x ). Mathematically, this projection may be represented bya linear projection operator P Λ , which maps the Hilbert space L (0 , b ) tospan { φ − Λ ( x ) , . . . , φ Λ ( x ) } , acting on (29). This mapping, however, causes aproblem in the non-linear term of (29), where by mode coupling the k -thFourier mode ( − Λ ≤ k ≤ Λ ) is influenced also by modes with | l | > Λ . This issuecan be resolved by choosing Λ large enough, for modes with h l ( t ) ∼ exp[ µ l t ], µ l from (28), (61), | l | > Λ will be damped out rapidly so that the bias in-troduced by limiting l to the interval R is small. Note that the restriction to h ∈ span { φ − Λ , . . . , φ Λ } also implies the introduction of restricted summationboundaries in the convolution term in (31), namely X l ∈ Z l ( k − l ) h l h k − l −→ X l ∈ R k \{ ,k } l ( k − l ) h l h k − l , k ∈ R , with R k defined by R k ≡ [max( − Λ, − Λ + k ) , min( Λ, Λ + k )] , k ∈ R . (48)This restriction to finitely many Fourier modes is not as harsh as it mightseem, since for very large wavenumbers the dynamics of the KPZ equation isgoverned by the Edwards-Wilkinson equation, which, due to its equilibriumbehavior, does not contribute to the thermodynamic uncertainty relation (24)(see e.g. [53,54,55,56]).With the cutoff Λ , condition (46) is of course fulfilled for α k = 1 ∀ k ∈ R and α k = 0 ∀ k / ∈ R . Inserting this choice of α k into (45) yields K ( x − x ′ ) = ∆ b " Λ X k =1 cos 2 πk x − x ′ b = ∆ δ ( x − x ′ ) (cid:12)(cid:12)(cid:12) span { φ − Λ ,...,φ Λ } . (49)Also, the choice of α k = 1 ∀ k ∈ R implies for the correlation function of theFourier coefficients η k ( t ) from (43) h η k ( t ) η l ( t ′ ) i = ∆ δ k, − l δ ( t − t ′ ) k, l ∈ R . (50)To end this section, a noise operator ˆ K describing spatial noise correlationswill be introduced as ˆ K ( · ) ≡ Z b dx ′ K ( x − x ′ )( · )( x ′ ) , (51)with kernel K ( x − x ′ ) from (49) and its inverse ˆ K − given byˆ K − ( · ) = Z b dx ′ K − ( x − x ′ )( · )( x ′ ) , (52)where its kernel reads K − ( x − x ′ ) = ∆ − δ ( x − x ′ ) (cid:12)(cid:12)(cid:12) span { φ − Λ ,...,φ Λ } . ield–Theoretic Thermodynamic Uncertainty Relation 13 λ eff , that replaces thecoupling constant λ from (25). To this end the following characteristic scalesare introduced, h = Hh s ; η = N η s ; x = bx s ; t = T t s . (53)Here H is a characteristic scale for the height field (not to be confused withthe notation for the Hilbert space), N a scale for the noise field, b is the char-acteristic length scale in space and T the time scale of the system. Choosingthe three respective scales according to H = r ∆ bν , N = r ∆ νb , T = b ν , (54)leads to the dimensionless KPZ equation on the interval x ∈ [0 , ∂ t s h s ( x s , t s ) = ∂ x s h s ( x s , t s ) + λ eff ∂ x s h s ( x s , t s )) + η s ( x s , t s ) , (55) h η s ( x s , t s ) i = 0 , (56) h η s ( x s , t s ) η s ( x ′ s , t ′ s ) i = K s ( x s − x ′ s ) δ ( t s − t ′ s ) . (57)Here, the effective dimensionless coupling constant is given by λ eff = λ ∆ / ν / b / , (58)and K s ( x s − x ′ s ) = 1 + 2 Λ X k =1 cos 2 πk ( x s − x ′ s ) . (59)The effective coupling constant λ eff is found in various works concerning theKPZ-Burgers equation; see e.g. [57,58,59,37].In the following sections we will perform all calculations for the dimension-less KPZ equation. This requires one simple adjustment in the linear differen-tial operator ˆ L on x s ∈ [0 , L s = ∂ x s , (60)with eigenvalues µ s, k = − π k (61)to the orthonormal eigenfunctions φ s, k ( x s ) = e πikx s . (62) Furthermore, the noise correlation function in Fourier space from (50) nowreads h η s, k ( t s ) η s, l ( t ′ s ) i = δ k, − l δ ( t s − t ′ s ) . (63)The scaling also effects the noise operators defined in (51), (52) at the end ofsubsection 3.2. The scaled ones readˆ K s ( · ) = Z dx ′ s K s ( x s − x ′ s )( · )( x ′ s ) , (64)and ˆ K − ( · ) = Z dx ′ s K − ( x s − x ′ s )( · )( x ′ s ) , (65)with K s ( x s − x ′ s ) from (59) and K − ( x s − x ′ s ) is defined via the integral-relation R dy s K s ( x s − y s ) K − ( y s − z s ) = δ ( x s − z s ).Note that for the sake of simplicity the subscript s will be dropped in thecalculations below where all quantities are understood as the scaled ones.3.4 Expansion in a Small Coupling ConstantReturning to the nonlinear integral equation of the k -th Fourier coefficient ofthe heights field, h k ( t ) from (31), now in its dimensionless form and with therestricted spectral range given by h k ( t ) = Z t dt ′ e µ k ( t − t ′ ) η k ( t ′ ) − π λ eff X l ∈ R k \{ ,k } l ( k − l ) h l ( t ′ ) h k − l ( t ′ ) , (66) k ∈ R , with { µ k } from (61), R k from (48) and all quantities dimensionless,an approximate solution will be constructed. Note, that the summation of thediscrete convolution in (66) is chosen such that it respects the above introducedcutoff in l as well as k − l , i.e. | l | , | k − l | ≤ Λ . For small values of the couplingconstant we expand the solution in powers of λ eff , i.e. h k ( t ) = h (0) k ( t ) + λ eff h (1) k ( t ) + λ h (2) k ( t ) + O ( λ ) , (67)with h (0) k ( t ) = Z t e µ k ( t − t ′ ) dW k ( t ′ ) , (68) h (1) k ( t ) = − π X l ∈ R k \{ ,k } l ( k − l ) Z t dt ′ e µ k ( t − t ′ ) h (0) l ( t ′ ) h (0) k − l ( t ′ ) , (69) h (2) k ( t ) = − π X l ∈ R k \{ ,k } l ( k − l ) Z t dt ′ e µ k ( t − t ′ ) × (cid:16) h (0) l ( t ′ ) h (1) k − l ( t ′ ) + h (1) l ( t ′ ) h (0) k − l ( t ′ ) (cid:17) (70) ield–Theoretic Thermodynamic Uncertainty Relation 15 Thus every h ( n ) k , n >
1, can be expressed in terms of h (0) m , m ∈ R , i.e. thestochastic convolution according to (32), which is known to be Gaussian.In the following calculations multipoint correlation functions have to be eval-uated, which can be simplified by Wick’s theorem, where a recurring termreads D h (0) k ( t ) h (0) l ( t ′ ) E . It is thus helpful to determine this correlation func-tion in general once and use this result later on. With (63) and k, l ∈ Z (andtherefore also for k, l ∈ R ) it follows that: D h (0) k ( t ) h (0) l ( t ′ ) E = e µ k t e µ l t ′ Z t dr Z t ′ ds e − µ k r e − µ l s h η k ( r ) η l ( s ) i = e µ k t e µ l t ′ δ k, − l − e − ( µ k + µ l )( t ∧ t ′ ) µ k + µ l = Π k,l ( t, t ′ ) δ k, − l , with Π k,l ( t, t ′ ) ≡ e µ k t e µ l t ′ − e − ( µ k + µ l )( t ∧ t ′ ) µ k + µ l . (71)Since for the auxiliary expression Π k,l the symmetries Π k,l ( t, t ′ ) = Π k, − l ( t, t ′ ) = Π − k,l ( t, t ′ ) = Π − k, − l ( t, t ′ ) (72)hold, it is found that D h (0) k ( t ) h (0) l ( t ′ ) E = D h (0) k ( t ) h (0) l ( t ′ ) E = Π k,l ( t, t ′ ) δ k, − l ; D h (0) k ( t ) h (0) l ( t ′ ) E = D h (0) k ( t ) h (0) l ( t ′ ) E = Π k,l ( t, t ′ ) δ k,l . (73) In this section we will show that the thermodynamic uncertainty relation from(24) holds for the KPZ equation driven by Gaussian white noise in the weak-coupling regime. In particular, the small- λ eff expansion from subsection 3.4will be employed.To recapitulate, the two ingredients needed for the thermodynamic uncertaintyrelation are (i) the long time behavior of the squared variation coefficient orprecision ǫ of Ψ g ( t ) from (9); (ii) the expectation value of the total entropyproduction in the steady state, h ∆s tot i from (22).4.1 Expectation and Variance for the Height FieldWith (7) adapted to the KPZ equation, namely Ψ g ( t ) = Z dx h ( x, t ) g ( x ) , (74) with g ( x ) as any real-valued L -function fulfilling R dxg ( x ) = 0, i.e. g ( x )possessing non-zero mean, we rewrite the variance as D ( Ψ g ( t ) − h Ψ g ( t ) i ) E = D ( Ψ g ( t )) E − h Ψ g ( t ) i . (75)As is shown below, ǫ can be evaluated for arbitrary time t >
0. However, thefinal interest is on the non-equilibrium steady state of the system. Therefore,the long-time asymptotics will be studied.
Evaluation of Expectation and Variance
In the small λ eff expansion, the expec-tation of the output Ψ g ( t ) from (74), with h ( x, t ) solution of the dimensionlessKPZ equation (55) to (57) reads: h Ψ g ( t ) i = X k,l ∈ R h h k ( t ) i g l (cid:0) e πikx , e πilx (cid:1) = λ eff X k ∈ R g k D h (1) k ( t ) E + O ( λ ) , (76)where g k and g k are the k -th Fourier coefficient of the weight function g ( x ) andits complex conjugate, respectively. Here the result from (68) is used as wellas the fact that odd moments of Gaussian random variables vanish identically.Replacing h (1) k ( t ) by the expression derived in (69) and using (73) leads to D h (1) k ( t ) E = − π e µ k t Z t dt ′ e − µ k t ′ X l ∈ R k \{ ,k } l ( k − l ) D h (0) l ( t ′ ) h (0) k − l ( t ′ ) E = − π X l ∈ R k \{ ,k } l ( k − l ) (cid:20) e ( µ l + µ k − l ) t − e µ k t ( µ l + µ k − l )( µ l + µ k − l − µ k ) − e µ k t − µ l + µ k − l ) µ k (cid:21) δ ,k . (77)Note, that in the case of k = 0 the second term in the last line of (77) isevaluated in the limit µ k →
0, which yields t . Since the interest is on thesteady state behavior, the long-time asymptotics of the two expressions in(77) above is studied. So, eq. (76) yields h Ψ g ( t ) i ≃ π g λ eff X l ∈ R \{ } l − µ l ) t + O ( λ ) , for t ≫ , (78)where g k = g − k ∀ k as g ( x ) ∈ R . Using the explicit form of µ k from (61), theexpression in (78) can be simplified according to h Ψ g ( t ) i = g λ eff Λ t + O ( λ ) , for t ≫ , (79)with Λ from (47). Equivalently, the steady state current from (8) reads J g = g λ eff Λ + O ( λ ) . (80) ield–Theoretic Thermodynamic Uncertainty Relation 17 The first term of the variance as defined in (75) reads in the small- λ eff expan-sion D ( Ψ g ( t )) E = * X k,l ∈ R h k ( t ) g k h l ( t ) g l + = X k,l ∈ R g k g l hD h (0) k ( t ) h (0) l ( t ) E + λ (cid:16)D h (1) k ( t ) h (1) l ( t ) E + D h (0) k ( t ) h (2) l ( t ) E + D h (2) k ( t ) h (0) l ( t ) E(cid:17) + O ( λ ) i , (81)where moments proportional to λ eff (and λ ) vanish due to (68) and (69) asthe two-point correlation function D h (0) k h (1) l E and its complex conjugate areodd moments.In Appendix A, we present the rather technical derivation of D ( Ψ g ( t )) E ≃ g − π ) λ X l ∈ R \{ } l µ l t + g λ X k ∈ R (cid:12)(cid:12)(cid:12)D h (1) k ( t ) E(cid:12)(cid:12)(cid:12) + O ( λ ) for t ≫ . (82)Subtraction of (78) squared from (82) leads to D ( Ψ g ( t )) E − h Ψ g ( t ) i ≃ g − π ) λ X l ∈ R \{ } l µ l t + O ( λ ) , for t ≫ . (83)Again, with µ k from (61), the above expression in (83) can be reduced to D ( Ψ g ( t )) E − h Ψ g ( t ) i = g (cid:20) λ π H (2) Λ (cid:21) t + O ( λ ) . (84)Here H (2) Λ = P Λl =1 /l is the so-called generalized harmonic number, whichconverges to the Riemann zeta-function ζ (2) for Λ → ∞ . Using (13), eq. (84)yields the diffusivity D g , D g = g (cid:20) λ π H (2) Λ (cid:21) + O ( λ ) . (85)With (84) and (79) squared, the first constituent of the thermodynamic un-certainty relation, ǫ = Var[ Ψ g ( t )] / h Ψ g ( t ) i from (9), is given for large timesby ǫ ≃ λ / (8 π ) H (2) Λ λ Λ t . (86) Note, since ǫ ≈ / ( λ t ), the long time asymptotics of the second term hasto scale as h ∆s tot i ∼ λ t for the uncertainty relation to hold. Note further,that the result for the precision of the projected output Ψ g ( t ) in the NESS isindependent of the choice of g ( x ).4.2 Alternative Formulation of the PrecisionBefore we continue with the calculation of the total entropy production, wewould like to mention an intriguing observation. From the field-theoretic pointof view, it seems natural to define the precision ǫ as ǫ ≡ D k h ( x, t ) − h h ( x, t ) ik E kh h ( x, t ) ik . (87)This is due to the fact that the height field h ( x, t ) is at every time instancean element of the Hilbert-space L ([0 , h ( x, t ) and its expectation is measured by its L -norm. Also the expectation squared is in this framework given by the L -norm squared. At a cursory glance, the definitions in (87) and (9) seem to beincompatible. However, for the case of the above calculations of ǫ for the one-dimensional KPZ equation, it holds up to O ( λ ) in perturbation expansionthat D ( Ψ g ( t ) − h Ψ g ( t ) i ) E = g D k h ( x, t ) − h h ( x, t ) ik E for t ≫ , h Ψ g ( t ) i = g kh h ( x, t ) ik . (88)Thus, with (88), it is obvious that in terms of the perturbation expansionboth definitions of the precision, as in (9) and (87), respectively, are equiva-lent. Equation (88) can be verified by direct calculation along the same linesas above in this section. By studying these calculations it is found perturba-tively that the height field h ( x, t ) is spatially homogeneous, which is reflectedby h h k ( t ) h l ( t ) i ∼ δ k, − l (see (73)) for the correlation of its Fourier-coefficients.Further, the long-time behavior is solely determined by the largest eigenvalueof the differential diffusion operator ˆ L = ∂ x , namely by µ = 0 (see e.g. (78)and (83), the essential quantities for deriving (88)).In the following, we would like to give some reasoning why the above two state-ments should also hold for a broad class of field-theoretic Langevin equations asin (1). For simplicity, we restrict ourselves in (1) to the case of one-dimensionalscalar fields Φ ( x, t ) and F [ Φ ( x, t )] = ˆ LΦ ( x, t ) + ˆ N [ Φ ( x, t )]. Here ˆ L denotes alinear differential operator and ˆ N a non-linear (e.g. quadratic) operator. ˆ L should be selfadjoint and possess a pure point spectrum with all eigenvalues µ k ≤ L = ( − p +1 ∂ px , p ∈ N , i.e. an arbitrary diffusion operator subjectto periodic boundary conditions). For this class of operators ˆ L there exists acomplete orthonormal system of corresponding eigenfunctions { φ k } in L ( Ω ).If it is further known, that the solution Φ ( x, t ) of (1) belongs at every time ield–Theoretic Thermodynamic Uncertainty Relation 19 t to L ( Ω ), we can calculate e.g. the second moment of the projected output Ψ g ( t ) according to (cid:10) ( Ψ g ( t )) (cid:11) = (cid:10) ( R Ω dx Φ ( x, t ) g ( x )) (cid:11) , where g ( x ) ∈ L ( Ω ) aswell. As is the case in e.g. equation (81), the second moment is determined bythe Fourier-coefficients Φ k ( t ) of Φ ( x, t ) and g k of g ( x ), namely D ( Ψ g ( t )) E = X k,l g k g l h Φ k ( t ) Φ l ( t ) i . (89)Like the KPZ equation, (1) is driven by spatially homogeneous Gaussian whitenoise η ( x, t ) with two-point correlations of the Fourier-coefficients η k ( t ) givenby h η k ( t ) η l ( t ) i ∼ δ k, − l . Therefore, we expect the solution to (1) subject toperiodic boundary conditions to be spatially homogeneous as well, at least inthe steady state, which implies h Φ k ( t ) Φ l ( t ) i ∼ δ k, − l , (90)see e.g. [60,61]. Hence, with (90), the expression in (89) becomes D ( Ψ g ( t )) E = g D ( Φ ( t )) E + X k =0 | g k | h Φ k ( t ) Φ − k ( t ) i . (91)Comparing (91) to D k Φ ( x, t ) k E , which is given by D k Φ ( x, t ) k E = X k h Φ k ( t ) Φ − k ( t ) i = D ( Φ ( t )) E + X k =0 h Φ k ( t ) Φ − k ( t ) i , (92)we find in the NESS D ( Ψ g ( t )) E ≃ g D k Φ ( x, t ) k E , (93)provided that the long-time behavior is dominated by the Fourier-mode withlargest eigenvalue, i.e. k = 0 with µ = 0. Under the same condition, the firstmoment of the projected output reads in the NESS h Ψ g ( t ) i = X k g k h Φ k ( t ) i ≃ g h Φ ( t ) i , (94)and thus ( h Ψ g ( t ) i ) ≃ g ( h Φ ( t ) i ) . (95)Similarly, kh Φ ( x, t ) ik = X k |h Φ k ( t ) i| ≃ ( h Φ ( t ) i ) for t ≫ , (96)which implies ( h Ψ g ( t ) i ) ≃ g kh Φ ( x, t ) ik . (97)Note, that g and Φ ( t ) have to be real throughout the argument (which isindeed the case for expansions with respect to the eigenfunctions of the generaldiffusion operators ˆ L from above). Hence, under the assumption that the prior mentioned requirements are met, which, of course, would have to be checkedfor every individual system (as was done in this section for the KPZ equation),the asymptotic equivalence in (93) and (97) validates the statement in (88)(and therefore, in the NESS, also (87)) for a whole class of one-dimensionalscalar SPDEs from (1).4.3 Total Entropy Production for the KPZ EquationThe total entropy production for the KPZ equation is obtained by inserting F γ [ h µ ( r , t )] = ∂ x h ( x, t ) + λ eff ( ∂ x h ( x, t )) and the explicit expression for theone-dimensional stationary probability distribution p s [ h ] into (22). The formof the latter is given in the following. The Fokker–Planck Equation and its 1D Stationary Solution
Let us brieflyrecapitulate the Fokker-Planck equation and its stationary solution in onespatial dimension for the KPZ equation.The Fokker-Planck equation corresponding to (55) for the functional proba-bility distribution p [ h ] reads, e.g. [62,40,63,31], ∂ p [ h ] ∂t = − Z dx δδh (cid:20)(cid:18) ∂ x h ( x, t ) + λ eff ∂ x h ( x, t )) (cid:19) p [ h ] − δp [ h ] δh (cid:21) = − Z dx δj [ h ] δh , (98) j [ h ] ≡ (cid:18) ∂ x h ( x, t ) + λ eff ∂ x h ( x, t )) (cid:19) p [ h ] − δp [ h ] δh , (99)with j [ h ] as a probability current.It is well known that for the case of pure Gaussian white noise, a stationarysolution, i.e. ∂ t p s [ h ] = 0, to the Fokker-Planck equation is given by [62,64,31] p s [ h ] ≡ exp h − k ∂ x h ( x, t ) k i . (100)This stationary solution is the same as the one for the linear case, namelyfor the Edwards-Wilkinson model. Note that in (100) we denote by k·k thestandard L -norm. Stationary Total Entropy Production
With (22), the total entropy productionin the NESS for the KPZ equation reads ∆s tot = ∆s m + s stat = 2 Z t dt ′ (cid:18) ˙ h, (cid:20) ∂ x h + λ eff ∂ x h ) (cid:21)(cid:19) − (cid:0) h, ∂ x h (cid:1) = (cid:20) Z t dt ′ (cid:16) ˙ h, ∂ x h (cid:17) − (cid:0) h, ∂ x h (cid:1) (cid:21) + λ eff Z t dt ′ (cid:16) ˙ h, ( ∂ x h ) (cid:17) . (101) ield–Theoretic Thermodynamic Uncertainty Relation 21 Using (cid:16) ˙ h, ∂ x h (cid:17) = ddt (cid:0) h, ∂ x h (cid:1) , and the initial condition h ( x,
0) = 0, thefirst term in (101) vanishes and thus ∆s tot = λ eff Z t dt ′ (cid:16) ˙ h ( x, t ′ ) , ( ∂ x h ( x, t ′ )) (cid:17) . (102)For Gaussian white noise, the expectation value of (102) is given by h ∆s tot i = λ eff Z t dt ′ D(cid:16) ˙ h ( x, t ′ ) , ( ∂ x h ( x, t ′ )) (cid:17) E = λ Z t dt ′ (cid:28)(cid:13)(cid:13)(cid:13) ( ∂ x h ( x, t ′ )) (cid:13)(cid:13)(cid:13) (cid:29) . (103)For a derivation of this result see Appendix B. Note that (103) and its deriva-tion remains true for h ∈ span { φ − Λ , . . . , φ Λ } . More generally, the expectationof the total entropy production may also be written as h ∆s tot i = λ Z t dt ′ D(cid:16) ( ∂ x h ( x, t ′ )) , ˆ K − ( ∂ x h ( x, t ′ )) (cid:17) E , (104)with ˆ K − from (65). Evaluating the Expectation of the Stationary Total Entropy Production
Above,an expression for the stationary total entropy production ∆s tot and its expec-tation value were derived (see eq. (103)). Inserting the Fourier representationfrom (26) and (62) into (103) leads to h ∆s tot i = (4 π ) λ Z t dt ′ Z dx X k ∈ R X m ∈ R e πix ( k − m ) × * X l ∈ R k \{ ,k } l ( k − l ) h l ( t ′ ) h k − l ( t ′ ) X n ∈ R m \{ ,m } n ( m − n ) h n ( t ′ ) h m − n ( t ′ ) + = (4 π ) λ Z t dt ′ X k ∈ R X l,n ∈ R k \{ ,k } l ( k − l ) n ( k − n ) × (cid:10) h l ( t ′ ) h k − l ( t ′ ) h n ( t ′ ) h k − n ( t ′ ) (cid:11) , (105)with R k from (48). As (105) above is already of order λ , it suffices to expandthe Fourier coefficients h i ( t ′ ) to zeroth order, which yields h ∆s tot i = (4 π ) λ Z t dt ′ X k ∈ R X l,n ∈ R k \{ ,k } l ( k − l ) n ( k − n ) × D h (0) l ( t ′ ) h (0) k − l ( t ′ ) h (0) n ( t ′ ) h (0) k − n ( t ′ ) E + O ( λ ) , (106) with h (0) i ( t ′ ) given by (68). Via a Wick contraction and using (73), the four-point correlation function in (106) reads D h (0) l ( t ′ ) h (0) k − l ( t ′ ) h (0) n ( t ′ ) h (0) k − n ( t ′ ) E = Π l,k − l ( t ′ , t ′ ) Π − n,n − k ( t ′ , t ′ ) δ ,k + Π l, − n ( t ′ , t ′ ) Π k − l,n − k ( t ′ , t ′ ) δ l,n + Π l,n − k ( t ′ , t ′ ) Π k − l, − n ( t ′ , t ′ ) δ n,k − l . (107)Inserting (107) into (106) leads to the following form of the total entropyproduction in the NESS, h ∆s tot i = (4 π ) λ X l,n ∈ R \{ } l n µ l µ n + 2 X k ∈ R X l ∈ R k \{ ,k } l ( k − l ) µ l µ k − l t + O ( λ ) . (108)Note that the long time behavior of h ∆s tot i is indeed of the form required, i.e. h ∆s tot i ∼ λ t (see remark after (86)), for the uncertainty relation to hold.With µ k from (61), the expression for the total entropy production from (108)reads h ∆s tot i = λ (cid:20) Λ + 3 Λ − Λ (cid:21) t + O ( λ ) . (109)Thus, with (23) and (109), the total entropy production rate reads σ = λ (cid:20) Λ + 3 Λ − Λ (cid:21) + O ( λ ) . (110)With (86) and (109), or, equivalently, (80), (85) and (110), the constituentsof the thermodynamic uncertainty relation are known. Hence, the productentering the TUR from (24) for the KPZ equation reads h ∆s tot i ǫ = 2 σ D g J g = 2 + (cid:18) − Λ (cid:19) + O ( λ ) . (111)Here, we deliberately refrain from writing h ∆s tot i ǫ = 5 − /Λ as this wouldsomewhat mask the physics causing this result. This point will be discussedfurther in the following. Edwards–Wilkinson Model for a Constant Driving Force
To give an interpre-tation of the two terms in (108) and consequently in (111), we believe it in-structive to briefly calculate the precision and total entropy production for thecase of the one-dimensional Edwards-Wilkinson model modified by an addi-tional constant non-random driving ‘force’ v and subject to periodic boundaryconditions. To be specific, we consider ∂ t h ( x, t ) = ∂ x h ( x, t ) + v + η ( x, t ) x ∈ [0 , , (112) ield–Theoretic Thermodynamic Uncertainty Relation 23 already in dimensionless form and with space-time white noise η . We denote(112) in the sequel with FEW for ‘forced Edwards-Wilkinson equation’. Fol-lowing the same procedure as described in section 3, we find the followingintegral equation for the k -th Fourier coefficient of the height field in FEW, h k ( t ) = e µ k t Z t dt ′ e − µ k t ′ [ v δ ,k + η k ( t ′ )] , (113)where again a flat initial configuration was assumed and µ k = − π k asabove. With (113), we get immediately in the NESS h Ψ g ( t ) i = g v t = J g t, (114)and thus h Ψ g ( t ) i = g v t as well as D ( Ψ g ( t )) E = X k,l ∈ R g k g l h h k ( t ) h l ( t ) i = X k,l ∈ R g k g l e ( µ k + µ l ) t Z t dr Z t ds e − µ k r − µ l s (cid:0) v δ ,k δ ,l + h η k ( r ) η l ( s ) i (cid:1) = g v t + X k ∈ R | g k | e µ k t − µ k = g v t + g t, for t ≫ . Thus, ǫ = D ( Ψ g ( t ) − h Ψ g ( t ) i ) E h Ψ g ( t ) i ≃ tv t = 1 v t . (115)As already discussed above in section 3, the Fokker-Planck equation corre-sponding to (112) has the stationary solution p s [ h ] = exp (cid:2) − R dx ( ∂ x h ) (cid:3) andthus, with (22) and (113), the total entropy production reads in the NESS h ∆s tot i = 2 Z dx h h ( x, t ) i v = 2 v t. (116)With (115) and (116), the TUR product for (112) is given by h ∆s tot i ǫ = 2 , (117)i.e. the thermodynamic uncertainty relation is indeed saturated for the Edwards-Wilkinson equation subject to a constant driving ‘force’ v . For the sake ofcompleteness we state the expressions for the current, diffusivity and rate ofentropy production in the non-equilibrium steady state, namely J FEW g = g v , D FEW g = g , σ FEW = 2 v . (118) With the calculations for FEW, we can now give an interpretation of thetwo terms in (109) and (111). The first term in squared brackets in (109)originates from the first term of (108), where the latter represents the actionof all higher-order Fourier modes on the mode k = 0 (see (107)). To illustratethis point further, observe that, in the NESS, we get according to (76) to (80)for the current: J g = 2 π g λ eff X l ∈ R \{ } l − µ l ) = g λ eff Λ, (119)and from the calculation above we see that it contains only the impact ofFourier modes l = 0 on the mode k = 0, which belongs to the constanteigenfunction φ ( x ) = 1. In other words, the modes l = 0 act like a constantexternal excitation, just in the same manner as v acts for FEW in (114).Comparing (119) to (114), we may set v = 2 π λ eff X l ∈ R \{ } l − µ l ) = λ eff Λ, (120)and get J g = g v in both cases.Following now the calculations for FEW, we would expect from (116) h ∆s tot i = 2 v t = (4 π ) λ X l ∈ R \{ } l − µ l ) t = λ Λ t, (121)which is in fact exactly the first term in the squared brackets from (108)and (109), respectively. Since with (120) also the expression for ǫ from (115)coincides with the first summand on the r.h.s. of (86), it is clear that bothcases result in the saturated TUR. This explains the value 2 on the r.h.s. of(111).Turning to the second term of (109), we see that it stems from the secondterm in (108). In contrast to the first term in (108), the second one does notonly measure the effect of the modes on the k = 0 mode but also on all othermodes k = 0. It further features interactions of the k and l modes among eachother via mode coupling. Hence, the mode coupling seems responsible for thelarger constant on the right hand side of (111), since by neglecting the modecoupling term in (109), the thermodynamic uncertainty relation was saturatedalso for the KPZ equation up to O ( λ ). To conclude this brief discussion, wegive the respective relations of the KPZ current (80), diffusivity (85) and totalentropy production rate (110) to FEW, namely J KPZ g = J FEW g + O ( λ ) ,D KPZ g = D FEW g + g λ π H (2) Λ + O ( λ ) ,σ KPZ = σ FEW + λ Λ − Λ O ( λ ) , (122) ield–Theoretic Thermodynamic Uncertainty Relation 25 with J FEW g , D FEW g and σ FEW from (118). We see that the additional modecoupling term in KPZ leads to corrections in D KPZ and σ KPZ of at leastsecond order in λ eff . For the case of λ eff → ∂ t h ( x, t ) = ∂ x h ( x, t ) + η ( x, t ), which possesses a genuine equilibrium steady state. Therefore, for thestandard EW we have J EW g = 0, σ EW = 0 and D EW g = g /
2. From (122) itfollows that for λ eff →
0, ( J g , σ, D g ) KPZ → ( J g , σ, D g ) FEW and from (118),(120) that ( J g , σ, D g ) FEW → ( J g , σ, D g ) EW = (0 , , g / J KPZ g and σ KPZ result solely from the KPZ non-linearity. Theimpact of the latter on the k = 0 Fourier mode (i.e. the spatially constantmode) results in contributions to J KPZ g and σ KPZ that can be modeled exactlyby FEW, the Edwards-Wilkinson equation driven by a constant force v from(112). We have introduced an analog of the TUR [1,2] in a general field-theoreticsetting (see (24)) and shown its validity for the Kardar-Parisi-Zhang equationup to second order of perturbation. To ensure convergence of the quantities en-tering the thermodynamic uncertainty relation for the case of Gaussian space-time white noise, we had to introduce an arbitrarily large but finite cutoff Λ of the corresponding Fourier spectrum. While this cutoff solves the issue ofdivergences, it naturally leads to subtleties in treating the non-linearity, as itsFourier spectrum is affected also by modes that are beyond the consideredspectral range. In order to minimize the resulting bias, the cutoff has to bechosen large enough such as to guarantee the dominance of the diffusive termover the non-linear term. To circumvent the introduction of a cutoff to ensureconvergence, a possible solution may be to induce a higher regularity by treat-ing spatially colored noise instead of Gaussian white noise and/or choosing ahigher order diffusion operator ˆ L (see e.g. [65,66]) . This is currently underinvestigation.As is obvious from (111), the field-theoretic version of the TUR for theKPZ equation displays a greater constant than the one in [1]. This is due tothe mode-coupling of the fields as a consequence of the KPZ non-linearity.To illustrate this point, we also treated the Edwards-Wilkinson equation insection 4.3, driven out of equilibrium by a constant velocity v , see (112). Byidentifying v with the influence of higher-order Fourier modes on the mode k = 0, we may interpret the first term in (108) as the contribution from theforced Edwards-Wilkinson equation, for which the TUR with constant equal to2 is saturated (see (117)), an observation which is in accordance with findingsin [8] for finite dimensional driven diffusive systems. The second term in (108)is the contribution to the entropy production made up by the interactionbetween Fourier modes of arbitrary order, which is due to the mode couplinggenerated by the KPZ non-linearity. It is this additional entropy productionthat weakens the dissipation bound in the TUR. Note, that also the first term in (108) is due to the mode-coupling, however is special in thus far that itmeasures only the impact of the other modes on the zeroth k -mode and doesnot include a response of the mode k = 0.Regarding future research, an intriguing topic is the question as to whetherthe findings in [8] concerning conditions for the saturation of the dissipationbound in the TUR for an overdamped two-dimensional Langevin equation canbe recovered in the present field-theoretic setting. Furthermore, it would beof great interest to employ the developed framework to other field-theoreticLangevin equations in order to observe the resulting dissipation bounds inthe corresponding TURs. Of special interest in this context is the stochasticBurgers equation, especially, if excited by a noise term suitable for generatinggenuine turbulent response (see [67]). A comparison of the predictions madein the present paper to numerical simulations of the KPZ equation seemsto be another intriguing task. Besides numerical calculations, it would alsobe of great interest to test our predictions via experimental realizations ofKPZ interfaces. Lastly, the formulation of a genuine non-perturbative, analyticformalism would also be of utmost interest. ield–Theoretic Thermodynamic Uncertainty Relation 27 A Evaluation of (81)
Using (73), the first term in (81) reads D h (0) k ( t ) h (0) l ( t ) E = Π k,l ( t, t ) δ k, − l = e ( µ k + µ l ) t − e − ( µ k + µ l ) t µ k + µ l δ k, − l . (123)Note that the case of k = 0 is treated like in (77). The second term in (81) is given by D h (1) k ( t ) h (1) l ( t ) E = (2 π ) X m ∈ R k \{ ,k } m ( k − m ) X n ∈ R l \{ ,l } n ( l − n ) Z t dt ′ e µ k ( t − t ′ ) Z t dr e µ l ( t − r ) × D h (0) m ( t ′ ) h (0) k − m ( t ′ ) h (0) n ( r ) h (0) l − n ( r ) E = − π ) X m ∈ R k \{ ,k } m ( k − m )( l + m ) Z t dt ′ e µ k ( t − t ′ ) Z t dr e µ l ( t − r ) × Π m,m ( t ′ , r ) Π k − m,l + m ( t ′ , r ) δ k, − l + (2 π ) X m ∈ R k \{ ,k } m ( k − m ) X n ∈ R l \{ ,l } n ( l − n ) Z t dt ′ e µ k ( t − t ′ ) Z t dr e µ l ( t − r ) × Π m,k − m ( t ′ , t ′ ) Π n,l − n ( r, r ) δ ,k δ ,l , (124)where we used Wick’s-theorem, (73) and (69). Note that the two Kronecker-deltas in thelast term of (124) can also be written as δ ,k δ ,l δ k, − l , such that the whole expression ismultiplied by δ k, − l . Again with Wick’s-theorem, (73) and (70) we can calculate the thirdand forth term of (81) accordingly and find D h (0) k ( t ) h (2) l ( t ) E = 4(2 π ) X m ∈ R l \{ ,l } ml ( l − m )( m − l ) Z t dt ′ e µ l ( t − t ′ ) Z t ′ dr e µ m ( t ′ − r ) × Π k,l ( t, r ) Π l − m,l − m ( t ′ , r ) δ k, − l , D h (2) k ( t ) h (0) l ( t ) E = 4(2 π ) X m ∈ R k \{ ,k } mk ( k − m )( m − k ) Z t dt ′ e µ k ( t − t ′ ) Z t ′ dr e µ m ( t ′ − r ) × Π k,l ( t, r ) Π k − m,k − m ( t ′ , r ) δ k, − l . (125)As can be seen from (123) to (125), all four terms in (81) contain a δ k, − l and thus (81)reduces to D ( Ψ g ( t )) E = X k ∈ R | g k | hD h (0) k ( t ) h (0) − k ( t ) E + λ (cid:16)D h (1) k ( t ) h (1) − k ( t ) E + D h (0) k ( t ) h (2) − k ( t ) E + D h (2) k ( t ) h (0) − k ( t ) E(cid:17) + O ( λ ) i (126)The first term of (126) is readily evaluated with (123) as D h (0) k ( t ) h (0) − k ( t ) E = Π k, − k ( t, t ) δ k,k = e µ k t − µ k = ( t for k = 0 , − µ k for k = 0 and t ≫ . (127)8 Oliver Niggemann, Udo SeifertThe second term of (126) reads with (124): D h (1) k ( t ) h (1) − k ( t ) E = 2(2 π ) X m ∈ R k \{ ,k } m ( k − m ) Z t dt ′ e µ k ( t − t ′ ) Z t dr e µ k ( t − r ) × Π m,m ( t ′ , r ) Π k − m,k − m ( t ′ , r )+ (2 π ) X m ∈ R \{ } m X n ∈ R \{ } n Z t dt ′ Π m,m ( t ′ , t ′ ) Z t dr Π n,n ( r, r ) (128)Hence, with D h (1) k ( t ) E from (77), the expression in (128) becomes D h (1) k ( t ) h (1) − k ( t ) E = 2(2 π ) e µ k t X m ∈ R k \{ ,k } m ( k − m ) µ m µ k − m Z t dt ′ Z t dr e ( µ m + µ k − m − µ k )( t ′ + r ) × (cid:16) − e − µ m ( t ′ ∧ r ) − e − µ k − m ( t ′ ∧ r ) + e − µ m + µ k − m )( t ′ ∧ r ) (cid:17) + D h (1) k ( t ) E . (129)Here the choice of the minimum of ( t ′ ∧ r ) is arbitrary, since for ( t ′ ∧ r ) = r the other case isobtained by simply interchanging r ↔ t ′ under the integral and vice versa; thus the resultsfor both choices are equivalent. In the following ( t ′ ∧ r ) = r is chosen. Hence, the integralexpression in (129) can be evaluated as e µ k t Z t dt ′ e ( µ m + µ k − m − µ k ) t ′ Z t ′ dr × (cid:16) e ( µ m + µ k − m − µ k ) r − e ( − µ m + µ k − m − µ k ) r − e ( µ m − µ k − m − µ k ) r + e − ( µ m + µ k − m + µ k ) r (cid:17) ≃ ( − t µ m for k = 0 µ k ( µ m + µ k − m + µ k ) for k = 0 for t ≫ . (130)Thus, with (129) and (130), the long time behavior of D h (1) k ( t ) h (1) − k ( t ) E is given by D h (1) k ( t ) h (1) − k ( t ) E ≃ D h (1) k ( t ) E + 2(2 π ) × (cid:20) − P l ∈ R \{ } l µ l (cid:21) t for k = 0 , P l ∈ R k \{ ,k } l ( k − l ) µ l µ k − l µ k ( µ l + µ k − l + µ k ) for k = 0 , (131)where we changed m → l . To save computational effort, rewrite the last two terms of (126)in the following way D h (0) k ( t ) h (2) − k ( t ) E + D h (2) k ( t ) h (0) − k ( t ) E = 2 Re (cid:20)(cid:28) h (0) k ( t ) h (2) k ( t ) (cid:29)(cid:21) . (132)Hence, it suffices to calculate one of the two expectation values. With (125) we see that (cid:28) h (0) k ( t ) h (2) k ( t ) (cid:29) = − π e µ k t Z t dt ′ e − µ k t ′ X m ∈ R k \{ ,k } km ( k − m ) e µ k − m t ′ Z t ′ dr e − µ k − m r × Π k,k ( t, r ) Π m,m ( t ′ , r ) , (133)ield–Theoretic Thermodynamic Uncertainty Relation 29where we substituted m → k − m and used the symmetry of Π k,l ( t, t ′ ) from (72). Note thatfor k = 0, the above expression in (133) vanishes. Thus in the following calculations k = 0is assumed. In this setting, (133) reads with (73) − π e µ k t Z t dt ′ e − µ k t ′ X l ∈ R k \{ ,k } kl ( k − l ) e µ k − l t ′ Z t ′ dr e − µ k − l r Π k,k ( t, r ) Π l,l ( t ′ , r )= − π X l ∈ R k \{ ,k } kl ( k − l )4 µ k µ l µ k ( µ l + µ k + µ k − l ) for k = 0 and t ≫ , (134)where we changed summation index m → l . Thus, with the results from (127), (131), (132)and (134), the expectation value of (81) reads in the long-time asymptotics D ( Ψ g ( t )) E ≃ g − π ) λ X l ∈ R \{ } l µ l t + g λ X k ∈ R D h (1) k ( t ) E + O ( λ ) . (135) B Expectation of the Total Entropy Production
The Fokker-Planck equation for the KPZ equation from (55) reads, like in section 4.3, ∂ t p [ h ] = − Z dx δδh (cid:20)(cid:18) ∂ x h + λ eff ∂ x h ) (cid:19) p [ h ] − δp [ h ] δh (cid:21) . (136)Due to the conservation of probability, there is a current j [ h ] given by j [ h ] = (cid:18) ∂ x h + λ eff ∂ x h ) (cid:19) p [ h ] − δp [ h ] δh = (cid:18) ∂ x h + λ eff ∂ x h ) (cid:19) p [ h ] − δ ln p [ h ] δh p [ h ]= (cid:20)(cid:18) ∂ x h + λ eff ∂ x h ) (cid:19) − δ ln p [ h ] δh (cid:21) p [ h ] (137) ≡ v [ h ] p [ h ] . (138)Following [41], expectation values of expressions like D ˙ h G [ h ] E are interpreted as D ˙ h G [ h ] E = h v [ h ] G [ h ] i = Z D [ h ] v [ h ] G [ h ] p [ h ] (139) ⇔ D ˙ h G [ h ] E = Z D [ h ] j [ h ] G [ h ] . (140)Since the goal is to find an expression for the expectation value of the total entropy produc-tion in the stationary state, ∆s tot , it is useful to choose p [ h ] as being the stationary solution p s [ h ] of the one-dimensional Fokker-Planck equation, which is given by p s [ h ] = exp h − k ∂ x h k i . (141)Inserting this in (137) yields for the stationary probability current j s [ h ] = λ eff ∂ x h ) p s [ h ] , (142)0 Oliver Niggemann, Udo Seifertwhere it was used that δp s [ h ] δh = 2 p s [ h ] ∂ x h. (143)Using the result from (140) and (142) leads to D ˙ h G [ h ] E = Z D [ h ] j s [ h ] G [ h ] = λ eff Z D [ h ] ( ∂ x h ) G [ h ] p s [ h ] = λ eff (cid:10) ( ∂ x h ) G [ h ] (cid:11) . (144)Here it is understood that h·i now denotes the expectation value with regard to the stationarydistribution p s [ h ].The total stationary entropy production ∆s tot is given by (see (102)) ∆s tot = λ eff Z t dt ′ Z dx ˙ h ( x, t ′ )( ∂ x h ( x, t ′ )) . (145)Hence its expectation value reads h ∆s tot i = λ eff Z t dt ′ Z dx D ˙ h ( x, t ′ )( ∂ x h ( x, t ′ )) E , (146)which is evaluated with the aid of (144): h ∆s tot i = λ Z t dt ′ (cid:28)(cid:13)(cid:13)(cid:13)(cid:0) ∂ x h ( x, t ′ ) (cid:1) (cid:13)(cid:13)(cid:13) (cid:29) . (147) C Regularity Results for the one-dimensional KPZ Equation
Dealing with the one-dimensional KPZ equation allows us to make use of the equivalence tothe stochastic Burgers equation and adapt the regularity results for the latter from [49,52,68,69,51]. In subsection 3.1 and subsection 3.2, we found that our operators ˆ L and ˆ K sharethe same set of eigenfunctions, which simplifies the results obtained by the authors of [52,51] to the following. Under the assumption that X k ∈ N k ρ − ( α B k ) < ∞ for some ρ > , (148)it is guaranteed almost surely that the solution u ( x, t ) of the one-dimensional noisy Burgersequation u ∈ C ([0 , T ] , H ), T >
0, with H = L ([0 , H = C ([0 , H to the solution u . Utilizing the mapping from KPZto Burgers via u ( x, t ) ≡ − ∂ x h ( x, t ), with h solution to the KPZ equation, which implies η B ( x, t ) = − ∂ x η KPZ ( x, t ) , (149)and therefore α B k ∼ k α KPZ k , (150)we get the following result for the 1d-KPZ equation: X k ∈ N k ǫ ( α KPZ k ) < ∞ ( ǫ = 2 ρ > ⇒ h ∈ C (cid:0) [0 , T ] , H ([0 , (cid:1) . (151)Here H ([0 , , f ∈ H ([0 , ⇔k f k L ([0 , < ∞ and k f ′ k L ([0 , < ∞ , where f ′ is understood as the weak derivative of f . It holds that H ([0 , ⊂ L ([0 , Λ of the Fourierspectrum instead of using our approximation to the KPZ solution as a spectral Galerkinscheme and letting Λ → ∞ . However, for the KPZ equation driven by spatially colored noisesatisfying (151) or even an adapted version of (151) to a higher order diffusion operator asield–Theoretic Thermodynamic Uncertainty Relation 31defined in subsection 3.1 (see e.g. [65,66]), in future work we want to derive a TUR takingthe full Fourier spectrum into account.We would like to conclude with the following remark. Since a couple of years, there exists acomplete existence and regularity theory for the KPZ equation driven by space-time whitenoise introduced by Hairer [70] (see also [71,72] and for further reading on the so-calledregularity structures developed in [70] see [73]). In [70] it is shown that the solutions ofthe KPZ equation with mollified noise converge after a suitable renormalization to thesolution of the renormalized KPZ equation with space-time white noise, when removing theregularization. It is due to this renormalization procedure (where a divergent quantity needsto be subtracted) and the poor regularity of the solution, that at present it is not obviousto us how the method developed in [70] can be of use for constructing a TUR. References
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