Large deviations of currents in diffusions with reflective boundaries
LLarge deviations of currents in diffusions withreflective boundaries
E. Mallmin , J. du Buisson , and H. Touchette SUPA, School of Physics and Astronomy, University of Edinburgh, Peter GuthrieTait Road, Edinburgh EH9 3FD, UK Institute of Theoretical Physics, Department of Physics, Stellenbosch University,Stellenbosch 7600, South Africa Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7600,South AfricaE-mail: [email protected], [email protected],[email protected]
10 February 2021
Abstract.
We study the large deviations of current-type observables defined forMarkov diffusion processes evolving in smooth bounded regions of R d with reflectionsat the boundaries. We derive for these the correct boundary conditions that mustbe imposed on the spectral problem associated with the scaled cumulant generatingfunction, which gives, by Legendre transform, the rate function characterizing thelikelihood of current fluctuations. Two methods for obtaining the boundary conditionsare presented, based on the diffusive limit of random walks and on the Feynman–Kac equation underlying the evolution of generating functions. Our results generalizerecent works on density-type observables, and are illustrated for an N -particle single-file diffusion on a ring, which can be mapped to a reflected N -dimensional diffusion.Keywords: dynamical large deviations, reflected diffusions, current fluctuations, single-file diffusion Submitted to:
J. Phys. A: Math. Theor.
1. Introduction
The main property of nonequilibrium systems that distinguishes them from equilibriumsystems is the existence of energy or particle currents produced by nonconservativeinternal or external forces, or coupling to reservoirs at different temperatures or chemicalpotentials [1]. These currents and their fluctuations have been the subjects of manystudies in the last decades, owing to their importance in biological transport phenomena[2, 3], and the existence of fundamental symmetries, referred to as fluctuation relations,which constrain the probability distribution of current-like quantities, such as theentropy production [4–6]. a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b arge deviations of currents in diffusions with reflective boundaries R d with reflection at the boundaries. Such processes areencountered in many applications, including in biology where they model the diffusionof nutrients within cells [14], and have been studied before in large deviation theory,though either in the low-noise limit [15–18] or for density-type observables defined astime integrals of the state of the process [19–26]. Here, we consider the long-time orergodic limit and focus on current-type observables, defined as integrals of the stateand increments of the reflected diffusion. For these, we show how the large deviationfunctions (viz., scaled cumulant generating function and rate function) characterizingthe fluctuations of observables can be obtained from a spectral equation that must besolved with special boundary conditions, taking into account the reflection of the processat the boundaries and the fact that the observable is current-like.These boundary conditions generalize those found recently for density-typeobservables [26] and are non-trivial in that they cannot be obtained by directly applyingthe duality argument used in [26]. This point is explained in the next section, followinga summary of large deviation theory as applied to dynamical currents defined in thelong-time limit. In the sections that follow, we then derive the boundary conditionsusing two different methods: one based on the diffusive limit of a random walk model,which has the advantage of being physically transparent, and another, more formalmethod based on the Feynman–Kac equation, which underlies the spectral equation,and a local-time formalism to describe the boundary behavior.A consequence of the boundary conditions imposed on the spectral problem is thatthe stationary current of the effective or driven process, introduced recently as a wayto understand how large deviations are created in time [27–30], vanishes in the normaldirection of the boundaries or walls limiting the diffusion. This is a natural resultgiven that the effective process also corresponds, in an asymptotic way, to the originalprocess conditioned on realizing a given fluctuation [30] and, therefore, should inheritthe zero-current condition of the original process resulting from the reflections.As an illustration of our results, we study the integrated particle current ofa heterogeneous single-file diffusion, consisting of N driven, non-identical particlesdiffusing on a ring without crossings [31]. This model can be mapped to a non-interactingdiffusion model in R N with a reflecting wall, for which the spectral problem can besolved exactly to obtain the scaled cumulant generating function and rate function ofthe current, characterizing its stationary value and fluctuations. Applications to other arge deviations of currents in diffusions with reflective boundaries
2. Dynamical large deviations with and without boundaries
We consider a d -dimensional Markov diffusion X ( t ) evolving in R d according to thestochastic differential equation (SDE)d X ( t ) = F ( X ( t )) + σ d W ( t ) , (1)where the drift vector F depends on the state of the process, whereas the noise matrix σ multiplying the Wiener noise is constant. We define the diffusion matrix D = σσ T .The process density ρ ( x , t ) = Prob[ X ( t ) = x | X (0) = x ] evolves according to theFokker-Planck (FP) equation ∂ t ρ ( x , t ) = ( L † ρ )( x , t ) , (2)expressed using the FP operator L † = −∇ · F + 12 ∇ · D ∇ , (3)which acts on the domain of twice-differentiable, integrable densities. We assume theexistence of a unique invariant density ρ ∗ for which L † ρ ∗ = 0. Time-dependent averagesof physical observables f of the process evolve in time according to ∂ t (cid:104) f ( X ( t )) (cid:105) = (cid:104) ( L f )( X ( t )) (cid:105) , (4)with the operator L = F · ∇ + 12 ∇ · D ∇ , (5)known as the Markov generator [32]. Using the standard inner product (cid:104) ρ, f (cid:105) = (cid:90) R d d x ρ ( x ) f ( x ) , (6)the Markov operators L and L † are related via the duality relation (cid:10) L † ρ, f (cid:11) = (cid:104) ρ, L f (cid:105) (7)for arbitrary densities ρ and functions f for which the inner product remains finite.We are concerned in this paper with time-integrated observables V T derived fromthe empirical current J T of the process. The empirical current at x is the number ofpasses through x (counted with sign) per unit time in a window [0 , T ]: J T ( x ) = 1 T (cid:90) T δ ( X ( t ) − x ) ◦ d X ( t ) (8a)= 1 T lim ∆ t → (cid:88) i δ (cid:18) X ( t i + ∆ t ) + X ( t i )2 − x (cid:19) ( X ( t i + ∆ t ) − X ( t i )) . (8b) arge deviations of currents in diffusions with reflective boundaries i indexes time-points in [0 , T ] separated by the vanishingduration ∆ t . The rationale for using the Stratonovich convention is that the current isproperly antisymmetric under time-reversal of trajectories and has an expectation equalin the long-time limit to the stationary current J F ,ρ ∗ = F ρ ∗ − D ∇ ρ. (9)From J T ( x ), it is possible to define other current-like dynamical observables as V T = (cid:90) R d g ( x ) · J T ( x ) d x = 1 T (cid:90) T g ( X ( t )) ◦ d X ( t ) , (10)where g is an arbitrary kernel vector field. The probability density P T ( v ) = Prob[ V T = v ]of such observables generally satisfies a large deviation principle for large observationtimes T , meaning that P T ( v ) = exp[ − T I ( v ) + o ( T )] . (11)The rate function I ( v ) characterizes the exponential decay of fluctuations v , that is,sustained deviations of the observable from the typical value(s) ¯ v for which I (¯ v ) = 0.Dynamical large deviation theory is concerned with calculating the rate function andwith describing via an effective process the subset of process realizations that give rise toany given fluctuation. Below we outline the basic elements of dynamical large deviationtheory, referring to [30] for details and derivations.In order to calculate the rate function, we introduce the scaled cumulant generatingfunction (SCGF) λ ( k ) = lim T →∞ T ln (cid:104) exp[ T kV T ] (cid:105) , (12)which is related to the rate function via Legendre–Fenchel (LF) transform, I ( v ) = sup k { kv − λ ( k ) } . (13)This assumes that I ( v ) is convex, which is guaranteed, for instance, if λ ( k ) iscontinuously differentiable in k [9]. Furthermore, it can be shown (via the Feynman–Kacformula [13]) that the generating function u ( x , t ) = (cid:104) exp[ tkV t ] (cid:105) x , (14)where the subscript x indicates the initial condition X (0) = x , has a semigroup structureand evolves in time according to ∂ t u ( x , t ) = L k u ( x , t ) , (15)where the ‘tilted’ generator L k is given by L k = F · ( ∇ + k g ) + 12 ( ∇ + k g ) · D ( ∇ + k g ) . (16) arge deviations of currents in diffusions with reflective boundaries λ ( i ) k and eigenfunctions r ( i ) k of L k : u ( x , t ) = (cid:88) i a i e λ ( i ) k t r ( i ) k ( x ) . (17)Inserting this decomposition in the definition of the SCGF, we find in general thatthe SCGF is the dominant (Perron–Frobenius) eigenvalue of L k , so we can write as ashorthand L k r k = λ ( k ) r k , (18) r k being the eigenfunction associated with the dominant eigenvalue and the SCGF λ ( k ) = max i Re λ ( i ) k .These spectral elements are also used in large deviation theory to define a newMarkov diffusion, called the effective or driven process, with generator [30] L eff k = r − k L k r k − λ ( k ) , (19)whose invariant density is ρ ∗ k ( x ) = (cid:96) k ( x ) r k ( x ) , (20)where (cid:96) k solves the dual eigenvalue problem L † k (cid:96) k = λ ( k ) (cid:96) k (21)with natural boundary conditions on R d . Here, L † k is related to L k via the duality (7)and is given explicitly by L † k = − ( ∇ − k g ) · F + 12 ( ∇ − k g ) · D ( ∇ − k g ) . (22)Note that the boundary conditions on r k are defined indirectly by imposing naturalboundary conditions for the density ρ ∗ k on R d .The effective process corresponds to a process with the same diffusion matrix asthe original one, but with a modified drift F k = F + D ( k g + ∇ ln r k ) . (23)When k is tuned according to k ( v ) = I (cid:48) ( v ) , (24)that is, the maximizer in (13), then the effective process gives v as the long-time valueof V T . In this way, this process can be interpreted as describing (asymptotically) theset of trajectories of the original process conditioned on V T = v (see [30] for precisestatement).An analogy with equilibrium statistical mechanics can be established by noting thatthe tilted generator corresponds (asymptotically) to the generator of a non-conservativeprocess constructed by penalizing the probability of every trajectory X over [0 , T ] by arge deviations of currents in diffusions with reflective boundaries T kV T ] as T → ∞ . This penalized distribution on trajectories isanalogous to the ‘canonical ensemble’, bar the missing normalization [30]. Followingthis analogy, the SCGF can be seen as being analogous to a free energy density, whilethe rate function, obtained via the LF transform, is analogous to an entropy density [9].The tuning (24) relates k to the inverse temperature in the canonical ensemble. We now consider a domain Ω ⊂ R d which has a smooth boundary ∂ Ω. In the interiorof the domain, the process X ( t ) is still described by the SDE (1). Consequently, theoperators L and L † are still given by (5) and (3), but must be supplemented withboundary conditions on the functions f and ρ on which they act, in order to accountfor the boundary behaviour.These boundary conditions are related through the duality relation (7), but withan inner product integrating over Ω rather than over all of R d . Starting from (cid:104)L ρ, f (cid:105) and performing repeated integration by parts to shift the derivatives from f to ρ (seeAppendix A), one finds (cid:104) ρ, L f (cid:105) = (cid:10) L † ρ, f (cid:11) − (cid:90) ∂ Ω d x f ( x ) J F ,ρ ( x ) · ˆ n ( x ) − (cid:90) ∂ Ω d x ρ ( x ) D ∇ f ( x ) · ˆ n ( x ) , (25)where ˆ n is the (inward) normal of ∂ Ω and the probability current J F ,ρ is as defined in(9). In order for the operators L and L † to be well-defined, independently of anyparticular ρ or f , it is necessary that the surface integral terms in (25) always vanish.A particular prescription that accomplishes this constitutes a boundary condition, andamounts to a restriction of the domains of the Markov operators. Note that if weput f ≡
1, then the vanishing of the boundary term corresponds to conservation ofprobability as it represents the zero net current through the boundary.In this paper, we are concerned with reflective boundaries. At the level of the FPequation, this means that the probability flow through the boundary vanishes at everypoint on the surface: J F ,ρ ( x ) · ˆ n ( x ) = 0 for all x ∈ ∂ Ω . (26)From (25), we note that (26) implies that the boundary condition on f is D ∇ f ( x ) · ˆ n ( x ) = 0 for all x ∈ ∂ Ω , (27)which is equivalent to ∇ f ( x ) · D ˆ n ( x ) = 0 for all x ∈ ∂ Ω , (28)since D is symmetric by definition. Thus, unlike the current, ∇ f does not vanish in thedirection normal to the surface, but in the direction D ˆ n , which is called the conormaldirection [14, 33]. arge deviations of currents in diffusions with reflective boundaries L k and (21) for its dual L † k , but now with boundaryconditions for r k and (cid:96) k on ∂ Ω in some way determined by the reflective boundaryof the original process. Clearly, these boundary conditions must be consistent with(26) and (27) for k = 0. In addition to this constraint, the duality relation for theeigenvectors should also hold for all k . Performing repeated integration by parts we find(Appendix B) (cid:104) (cid:96) k , L k r k (cid:105) = (cid:68) L † k (cid:96) k , r k (cid:69) − (cid:90) ∂ Ω d x J F k ,(cid:96) k r k ( x ) · ˆ n ( x ) , (29)where F k is the modified drift (23). Interpreting again (cid:96) k r k = ρ ∗ k as the stationarydensity of an effective process with drift F k , the vanishing of the boundary term in (29)expresses the conservation of probability for the effective process.On physical grounds, it is reasonable to suppose that if the original process hasreflective boundaries, then so does the effective process, meaning that the boundaryterm in (29) vanishes because J F k ,(cid:96) k r k ( x ) · ˆ n ( x ) = 0 for all x ∈ ∂ Ω . (30)However, as we are about to see, this condition does not allow us to uniquely determineboundary conditions for r k and (cid:96) k separately. Indeed, we can write, for arbitraryconstant c , J F k ,(cid:96) k r k = ( F + k D g ) (cid:96) k r k + 12 D (cid:96) k ∇ r k − D r k ∇ (cid:96) k (31a)= (cid:20) [ F + (1 − c ) k D g ] (cid:96) k − D ∇ (cid:96) k (cid:21) r k + (cid:96) k (cid:20) ck D g r k + 12 D ∇ r k (cid:21) . (31b)This identity can be separated into the boundary conditions (cid:26) [ F ( x ) + (1 − c ) k D g ( x )] (cid:96) k ( x ) − D ∇ (cid:96) k ( x ) (cid:27) · ˆ n ( x ) = 0 , (32a) (cid:26) ck D g r k + 12 D ∇ r k (cid:27) · ˆ n ( x ) = 0 , (32b)for x ∈ ∂ Ω and any c . The ambiguity arises from the fact that any boundary termproportional to (cid:96) k r k that vanishes as k → arge deviations of currents in diffusions with reflective boundaries
8a conditioned lattice problem with boundary (Sec. 3), and the other based on consideringlocal time at the boundary in the Feynman–Kac formula that defines the tilted generator(Sec. 4). From both approaches, the boundary conditions emerge as (cid:8) F ( x ) (cid:96) k ( x ) − D ( ∇ − k g ) (cid:96) k ( x ) (cid:9) · ˆ n ( x ) = 0 , (33a) D ( ∇ + k g ( x )) r k ( x ) · ˆ n ( x ) = 0 . (33b)These boundary conditions correspond to (32) with c = 1 / ∇ → ∇ + k g for r k and −∇ → −∇ + k g for (cid:96) k .Furthermore (33b) implies that, on the boundary, the normal component of the effectivedrift coincides with the original drift: F k ( x ) · ˆ n ( x ) = F ( x ) · ˆ n ( x ) for all x ∈ ∂ Ω . (34)This situation was shown to hold also for the large deviations of density-like observablesin the presence of a reflecting boundary [26].Finally, we remark that in the special case of an observable satisfying D g ( x ) · ˆ n ( x ) = 0 for all x ∈ ∂ Ω , (35)the tilted boundary conditions (32) take the same form as the original ones, i.e., (26)and (27), and are independent of c . The condition (35) is a necessary condition for thetilted generators to be symmetrizable.
3. Boundary conditions from the diffusive limit
A strategy to derive the correct boundary conditions on r k and (cid:96) k is to consider theoriginal diffusion as the limit of a jump process [34]; that is, to set up a sequence of jumpprocess N ( t ) on a lattice structure L , parametrized by the site separation a , togetherwith a lattice-current observable A T . The transition rates of the jump process are takento scale with a such that a diffusive limit exists, giving as a → L → Ω, N ( t ) → X ( t ),and A T /a → V T . Then also the spectral elements associated with the conditioning on A T map from lattice to continuum, giving in this limit bulk and boundary equationsfor (cid:96) k and r k .For simplicity, we suppose L to be a cubic lattice with a planar boundary. Since theboundary ∂ Ω converged to is always locally planar (because it is smooth), this is not anessential limitation. The process N ( t ) evolving on L is then a random walk, as definedin Fig. 1. For each i labelling a spatial axis, the hopping rate is p i ( n ) forwards and q i ( n ) backwards. The hopping rate into a boundary vanishes, which can be interpretedas reflection, as explained in Fig. 1(c). arge deviations of currents in diffusions with reflective boundaries A T = (cid:88) n , n (cid:48) ∈ L α ( n (cid:48) | n ) C T ( n (cid:48) | n ) , (36)where the empirical flow C T counts the number of transitions over the specified bond: C T ( n (cid:48) | n ) = 1 T (cid:88) t ∈ [0 ,T ]: N ( t + ) (cid:54) = N ( t − ) δ N ( t − ) , n δ N ( t + ) , n (cid:48) . (37)Because we seek to map A T onto the diffusion observable V T , we choose an antisymmetric α : α ( n (cid:48) | n ) = − α ( n | n (cid:48) ) . (38)We can then write (36) as A T = (cid:88) n (cid:88) i α ( n + ˆ e i | n ) J T ( n + ˆ e i | n ) (39)where the ˆ e i is the single-site translation vector for axis i and J T ( n (cid:48) | n ) is the empiricallattice current, defined in terms of the empirical flow as J T ( n (cid:48) | n ) = C T ( n (cid:48) | n ) − C T ( n | n (cid:48) ) . (40)The contraction of the current in (39) giving A T is the jump process analog of (10) forthe diffusion, with α playing the role of g .As for diffusions, the SCGF of A T corresponds to a dominant eigenvalue, this timeof a | L | × | L | matrix L s with elements [30] L s ( n (cid:48) , n ) = W ( n | n (cid:48) ) e sα ( n | n (cid:48) ) − δ n , n (cid:48) (cid:88) n (cid:48)(cid:48) W ( n (cid:48)(cid:48) | n ) . (41)The non-zero transition rates in this expression are W ( n + ˆ e i | n ) = p i ( n ) , n + ˆ e i ∈ L , (42a) W ( n − ˆ e i | n ) = q i ( n ) , n − ˆ e i ∈ L . (42b) The diffusive limit relating the master equation of N ( t ) to the FP equation of X ( t ) isdefined by the following scaling relations [34]: F i ( x ) = lim a → a ( p i ( n ) − q i ( n )) (43a) σ i = lim a → a ( p i ( n ) + q i ( n )) , (43b) arge deviations of currents in diffusions with reflective boundaries p p q q (a) p q q (b) p q (c) Figure 1.
Illustrations of transitions available for the random walker (a) away fromthe boundary; (b) next to the boundary. (c) To interpret the behavior in (b) as aphysical reflection, we may put the wall half a site away from the last inhabitable site,and allow hops of length 1 / / where points in Ω relate to points in L as x = a n . Equivalently, we may state p i ( n ) = σ a + F i ( x )2 a + O (1) , (44a) q i ( n ) = σ a − F i ( x )2 a + O (1) . (44b)To show that A T /a converges to V T in this limit, we first use the fact that α isantisymmetric to write α ( n ± ˆ e i | n ) /a d = G ( x + a ˆ e i ) − G ( x ) (45a)= ± ag i ( x ) + a ∂ x i g i ( x ) + O ( a ) (45b)where G is an arbitrary smooth function independent of a such that g i = ∂ x i G .Next, we note that the lattice empirical current over the n → n + ˆ e i bond is J T ( n + ˆ e i | n ; N ) = 1 T (cid:88) t (cid:2) δ N ( t − ) , n δ N ( t + ) , n +ˆ e i − δ N ( t − ) ,n +1 δ N ( t + ) ,n (cid:3) (46a)= 1 T (cid:88) t (cid:2) δ N ( t − ) , n δ N ( t + ) , n +ˆ e i + δ N ( t − ) , n +ˆ e i δ N ( t + ) ,n (cid:3) ( N i ( t + ) − N i ( t − )) , (46b)where the second line follows because N i ( t + ) and N i ( t − ) differ by precisely one step.We now discretize time into points t j narrowly separated by intervals ∆ t ( a ), such thatthe jump process makes at most one jump in each interval for any value of the siteseparation a . For any such trajectory, δ N ( t j ) ,n δ N ( t j +∆ t ) , n +ˆ e i + δ N ( t j ) , n +ˆ e i δ N ( t j +∆ t ) , n = δ N ( tj )+ N ( tj +∆ t )2 , n + ˆ e i , (47)which is seen from the fact that both sides are symmetric under exchange of N ( t − ) and N ( t + ). In the diffusive limit we replace N ( t ) = X ( t ) /a , δ n , n (cid:48) = δ ( x /a − x (cid:48) /a ), and thus arge deviations of currents in diffusions with reflective boundaries J T ( n + ˆ e i | n ) = 1 T (cid:88) j δ N ( tj )+ N ( tj +∆ t )2 , n + ˆ e i ( N i ( t j + ∆ t ) − N i ( t j )) (48a)= 1 T (cid:88) j δ (cid:18) X ( t j + ∆ t ) + X ( t j )2 − x (cid:19) ( X i ( t j + ∆ t ) − X i ( t j )) (48b) a → = ˆ e i · J T ( x ) , (48c)as defined in (8). Hence A T /a = (cid:88) n (cid:88) i α ( n + ˆ e i | n ) J ( n + ˆ e i | n ) /a (49a)= (cid:90) d x a d (cid:88) i a d +1 g i ( x )ˆ e i · J T ( x ) /a + O ( a ) (49b) a → = V T . (49c) We now derive the diffusive limit of the spectral elements, assuming the followingdiffusive scaling between the lattice and continuum elements: s = a d k (50a)Λ s = λ ( k ) + O ( a ) (50b) L s ( n ) = a d (cid:96) k ( x ) + O ( a d +1 ) (50c) R s ( n ) = a d r k ( x ) + O ( a d +1 ) (50d)To begin, we consider the limit of the right eigenvalue equation,Λ s R s ( n ) = (cid:88) n (cid:48) L s ( n , n (cid:48) ) R s ( n (cid:48) ) . (51)For n away from the boundary sites,Λ s R s ( n ) = (cid:88) i (cid:104) p i ( n ) e s α ( n +ˆ e i | n ) R s ( n + ˆ e i ) + q i ( n ) e s α ( n − ˆ e i | n ) R s ( n − ˆ e i ) − ( p i + q i )( n ) R s ( n ) (cid:105) . (52) arge deviations of currents in diffusions with reflective boundaries a , and suppressing the function arguments x and n , λ ( k ) r k = (cid:88) i (cid:20) p i (cid:18) kag i + 12 ka ∂ x i g i + k a g i (cid:19) (cid:18) r k + a∂ x i r k + 12 ∂ x i r k (cid:19) + q i (cid:18) − kag i + 12 k a ∂ x i g i + k a g i (cid:19) (cid:18) r k − a∂ x i r k + 12 ∂ x i r k (cid:19) − ( p i + q i ) r k (cid:21) (53a)= (cid:88) i = (cid:20) a ( p i − q i ) ( ∂ x i r k + kg i r k ) + a p i + q i ) × (cid:0) ∂ x i r k + r k ∂ x i g i + 2 kg i ∂ x i r k + k g i r k (cid:1) (cid:21) (53b)= (cid:88) i (cid:26) F i ( ∂ x i + kg i ) r k + ( ∂ x i + kg i ) σ i ∂ x i + kg i ) r k (cid:27) (53c)= L k r k , (53d)with L k as in (16). This recovers the spectral equation for r k in the bulk.Now let us take n to be a boundary site as in Fig. 1(b). ThenΛ s R s ( n ) = q ( n ) e s α ( n − ˆ e | n ) R s ( n − ˆ e ) − q ( n ) R s ( n )+ (cid:88) i> (cid:104) p i ( n ) e s α ( n +ˆ e i | n ) R s ( n + ˆ e i ) + q i ( n ) e s α ( n − ˆ e i | n ) R s ( n − ˆ e i ) − ( p i + q i )( n ) R s ( n ) (cid:105) . (54)Thus, including all relevant orders, λ ( k ) r k = q (1 − kag ) ( r k − a∂ x r k ) − q r k + O (1) , (55)where we have used (53) to neglect the sum on the i > O (1 /a ). Substituting q i with (44b), multiplying both sides by a , andtaking a →
0, we then arrive at 0 = 12 σ ( ∂ x r k + g r k ) , (56)which generalizes, including all other components, to D ( ∇ + k g ) r k · ˆ n = 0 . (57)Thus, we find the boundary condition (33b), corresponding to (32) with c = 1 / r k has been established, the boundarycondition (33a) for (cid:96) k follows uniquely from duality. One may also verify that thisboundary condition follows from the diffusive limit of the left eigenvalue equation, in acalculation analogous to that of r k . Furthermore, the duality relation (29) is the resultof applying the diffusive limit to the trivial identity (cid:88) n , n (cid:48) L s ( n ) L s ( n , n (cid:48) ) R s ( n (cid:48) ) = (cid:88) n , n (cid:48) R s ( n ) L (cid:62) s ( n , n (cid:48) ) L s ( n (cid:48) ) . (58) arge deviations of currents in diffusions with reflective boundaries
4. Boundary conditions from Feynman-Kac expectation
We provide in this section an alternative derivation of the boundary condition (33b) on r k , proceeding directly from the generating function u ( x , t ), as defined in (14), which,from its spectral decomposition (17), shares the boundary conditions placed on theeigenfunctions of L k . To account for the reflection upon reaching the boundary ∂ Ω, weemploy a formulation of reflected SDEs, introduced by Skorokhod [35] and Tanaka [36],based on the following modified SDE:d X ( t ) = F ( X ( t )) dt + σ d W ( t ) + ˆ γ ( X ( t )) d L ( t ) . (59)The first two terms on the right-hand side describe the evolution of X ( t ) inside thedomain Ω, whereas the last term pushes the process inside Ω in the direction of the unitvector ˆ γ ( x ) in the event that the process reaches x ∈ ∂ Ω. The extra random process L ( t ) accounting for the reflection is called the local time, since it is incremented onlywhen the process reaches ∂ Ω, and is known [37] to be such that (cid:104) d L ( t ) (cid:105) X ( t )= x = O ( √ dt ) . (60)Therefore, the increment d X ( t ) on x ∈ ∂ Ω satisfies (cid:104) d X ( t ) (cid:105) X ( t )= x = F ( x ) dt + σ (cid:104) d W ( t ) (cid:105) + ˆ γ ( x ) (cid:104) d L ( t ) (cid:105) X ( t )= x = ˆ γ ( x ) (cid:15) + O ( (cid:15) ) (61)where we have used the fact that (cid:104) d W ( t ) (cid:105) = for all t , and where (cid:15) = O ( √ dt ) suchthat dt ( (cid:15) ) = O ( (cid:15) ). Here, we take ˆ γ to be in the conormal direction, that is,ˆ γ ( x ) = D ˆ n ( x ) | D ˆ n ( x ) | . (62)This choice is necessary (see [38, Thm. 2.6.1]) to preserve the zero-current condition(26) associated with reflection.Our goal now is to understand the effect of the boundary dynamics on the generatingfunction u ( x , t ) of the observable V T , as defined in (14). To this end, we consider a point x ∈ ∂ Ω and write the generating function as u ( x , t ) = (cid:42) exp (cid:34) k (cid:90) dt ( (cid:15) )0 g ( X ( s )) ◦ d X ( s ) + (cid:90) tdt ( (cid:15) ) g ( X ( s )) ◦ d X ( s ) (cid:35)(cid:43) x , (63)having isolated in the first integral the contribution from the reflection, which takesplace over the infinitesimal time dt ( (cid:15) ). Using the Stratonovich discretization, as in (8),we have exp (cid:34) k (cid:90) dt ( (cid:15) )0 g ( X ( s )) ◦ d X ( s ) (cid:35) = exp (cid:20) k g (cid:18) x + d X (0)2 (cid:19) · d X (0) (cid:21) , (64)so that u ( x , t ) = (cid:28) exp (cid:20) k g (cid:18) x + d X (0)2 (cid:19) · d X (0) (cid:21) exp (cid:20)(cid:90) tdt ( (cid:15) ) g ( X ( s )) ◦ d X ( s ) (cid:21)(cid:29) x . (65) arge deviations of currents in diffusions with reflective boundaries u ( x , t ) = (cid:90) d(ˆ ξ δ ) p (cid:0) d X (0) = ˆ ξ δ (cid:12)(cid:12) X (0) = x (cid:1) e k g (cid:16) x + ˆ ξ δ (cid:17) · ˆ ξ δ (cid:28) exp (cid:20)(cid:90) tdt ( (cid:15) ) g ( X ( s )) ◦ d X ( s ) (cid:21)(cid:29) X ( dt )= x +ˆ ξ δ (66)using the conditional probability density of the first increment d X (0) from X (0) = x ∈ ∂ Ω , which includes all the information about the reflections on the boundary. Using thedefinition of the generating function for the last factor in the integral, we then obtain u ( x , t ) = (cid:90) d(ˆ ξ δ ) p (cid:0) d X (0) = ˆ ξ δ (cid:12)(cid:12) X (0) = x (cid:1) e k g (cid:16) x + ˆ ξ δ (cid:17) · ˆ ξ δ u ( x + ˆ ξ δ, t − dt ( (cid:15) )) . (67)At this point, we perform Taylor expansions in both space and time, starting withthe one in space, which gives to first order in δ : u ( x , t ) = (cid:90) d(ˆ ξ δ ) , p (cid:0) d X (0) = ˆ ξ δ (cid:12)(cid:12) X (0) = x (cid:1)(cid:20) u ( x , t − dt ( (cid:15) ))+ ∇ u ( x , t − dt ( (cid:15) )) · ˆ ξ δ + ku ( x , t − dt ( (cid:15) )) g ( x ) · ˆ ξ δ + O ( δ ) (cid:21) . (68)This becomes u ( x , t ) = u ( x , t − dt ( (cid:15) )) + ∇ u ( x , t − dt ( (cid:15) )) · (cid:104) d X (0) (cid:105) X (0)= x + ku ( x , t − dt ( (cid:15) )) g ( x ) · (cid:104) d X (0) (cid:105) X (0)= x + (cid:10) | d X (0) | (cid:11) X (0)= x , (69)given that (cid:90) d(ˆ ξ δ ) p (cid:0) d X (0) = ˆ ξ δ (cid:12)(cid:12) X (0) = x (cid:1) = 1 (70)and (cid:90) d(ˆ ξ δ ) p (cid:0) d X (0) = ˆ ξ δ (cid:12)(cid:12) X (0) = x (cid:1) ˆ ξ δ = (cid:104) d X (0) (cid:105) X (0)= x , (71)and noting that δ = | d X (0) | . Moreover, since x ∈ ∂ Ω, we have (cid:104) d X (0) (cid:105) X (0)= x = D ˆ n ( x ) | D ˆ n ( x ) | (cid:15) + O ( (cid:15) ) (72)from (61) and (62), yielding u ( x , t ) = u ( x , t − dt ( (cid:15) )) + ∇ u ( x , t − dt ( (cid:15) )) · D ˆ n ( x ) | D ˆ n ( x ) | (cid:15) + ku ( x , t − dt ( (cid:15) )) g ( x ) · D ˆ n ( x ) | D ˆ n ( x ) | (cid:15) + O ( (cid:15) ) . (73)Considering now the Taylor expansion in time, we have u ( x , t − dt ( (cid:15) )) = u ( x , t ) − L k dt ( (cid:15) ) u ( x , t ) + O ( dt ( (cid:15) ) ) = u ( x , t ) + O ( (cid:15) ) , (74) arge deviations of currents in diffusions with reflective boundaries dt ( (cid:15) ) = O ( (cid:15) ). Therefore, we find u ( x , t ) = u ( x , t ) + ∇ u ( x , t ) · D ˆ n ( x ) | D ˆ n ( x ) | (cid:15) + ku ( x , t ) g ( x ) · D ˆ n ( x ) | D ˆ n ( x ) | (cid:15) + O ( (cid:15) ) (75)that is, ∇ u ( x , t ) · D ˆ n ( x ) | D ˆ n ( x ) | = − ku ( x , t ) g ( x ) · D ˆ n ( x ) | D ˆ n ( x ) | (76)or D ∇ u ( x , t ) · ˆ x = − ku ( x , t ) D g ( x ) · ˆ n ( x ) , (77)using the fact that D is symmetric. This is the boundary condition satisfied by thegenerating function for all x ∈ ∂ Ω. The eigenfunctions r ( i ) k ( x ) of the tilted generator L k share the same boundary condition, so we have in the end D ∇ r k ( x ) · ˆ n ( x ) = − kr k ( x ) D g ( x ) · ˆ n ( x ) for all x ∈ ∂ Ω , (78)which reproduces the boundary condition (33b), obtained also from the diffusive limit.The boundary condition on (cid:96) k can then be found in the usual manner via the dualityrelation (29), thereby recovering (33a).The same calculation can be performed, in principle, for an arbitrary reflectiondirection ˆ γ ( x ), in which case the boundary condition becomes ∇ u ( x , t ) · ˆ γ ( x ) = − ku ( x , t ) g ( x ) · ˆ γ ( x ) . (79)Comparing with the duality relation (25) for k = 0 then shows that we only obtain thezero-current condition at the boundary when the chosen direction for reflection is theconormal direction, as mentioned before. It is also interesting to note that, if we repeatthe calculation for a density-type observable of the form A T = 1 T (cid:90) T f ( X ( s )) d s, (80)then the boundary condition is D ∇ r k ( x ) · ˆ n ( x ) = 0 , (81)which is the result obtained in [26] using only the duality relation.
5. Exactly solvable example: Heterogeneous single-file diffusion on a ring
We illustrate the formalism and results developed in the previous sections for a single-file diffusion model, based on an earlier lattice model [39, 40], which was recently solvedin [31] for the steady state. The model consists of N distinct point-particles coexistingon a ring S of circumference L , as illustrated in Fig. 2. Each particle i has a constantintrinsic velocity v i and a diffusivity D i = σ i arising from white noise of strength σ i . Theparticles interact through volume exclusion: if one particle attempts to overtake another, arge deviations of currents in diffusions with reflective boundaries L/ v i , D i (a) t xL T (b) Figure 2. (a) Illustration of heterogeneous single-file diffusion on a ring. (b) Space-time plot of a typical realization. Due to hardcore exclusion, i.e. reflection, theparticles’ paths do not cross. that move is reflected. Collecting the (stochastic) particle positions X i ( t ) taking valuesin S into a vector X ( t ) on a domain Ω ⊂ S N , we obtain an N -dimensional diffusion ofthe form (1) where the drift F is the collection v = ( v , . . . , v N ) (cid:62) of intrinsic velocities,and D is the diagonal matrix diag { D , . . . , D N } .The boundary ∂ Ω of the process consists of those configurations for which two (ormore) particles are immediately adjacent. The hardcore exclusion rule translates intothe reflective boundary condition (26). For two particles, for instance, the boundaryconsists of X = X , which in the space S (cid:39) [0 , L ) × [0 , L ) is a diagonal with normalˆ n = (ˆ e − ˆ e ) / √ − , +1) / √
2. Generalising to N particles, we then find that theboundary conditions areˆ e i · J v ,ρ ( x , t ) | x i = x j = ˆ e j · J v ,ρ ( x , t ) | x i = x j . (82)The periodicity of the ring is implemented by the condition ρ ( x , t ) = ρ ( x + L , t ) , (83)for the density, where is the vector of ones, so that L is the translation vector movingall particles simultaneously by one period L .It was shown in [31] that the invariant density of this process is ρ ∗ ( x ) ∝ exp[ k · x ] . (84)This result assumes that x ∈ [0 , L ) N and that the ordering of particles is consistentwith that of the initial condition, clearly conserved by the dynamics. Without loss ofgenerality, we assume . . . x i − < x i < x i +1 < . . . (modulo L ). The vector k has elements k i = v i D i − ˜ vD i , (85) arge deviations of currents in diffusions with reflective boundaries v satisfies ˜ v (cid:101) D = N (cid:88) i =1 v i D i , and 1 (cid:101) D = N (cid:88) i =1 D i . (86)It is clear from the geometrical constraints on the particles’ motion that they must havea common net velocity in the long-time limit, corresponding in fact to ˜ v . If follows from(84) that ˆ e i · J v ,ρ ∗ ( x ) = ˜ vρ ∗ ( x ) , (87)independent of i , which confirms the interpretation of ˜ v . Note that k · = 0, whichensures the periodicity (83).It is natural to consider as a current-like observable the empirical velocity of particle i , given by the i th component of the empirical current J T integrated over Ω. However,since all particles must have the same net velocity for long averaging periods, allobservables of the form (10) with g a constant vector whose components sum to one( · g = 1) should have the same large deviations. To validate this claim, we keep g arbitrary apart from these constraints, and thus consider the current observable V T = g · (cid:90) Ω J T ( x ) d x . (88)The empirical velocity of particle i is obtained by choosing g j = δ ij .To find the dominant eigenvalue λ ( k ) and eigenvector r k related to this observable,we consider as an ansatz r k ( x ) = exp[ a · x ] , (89)with a to be determined. We observe that · ˆ n ( x ) = 0 for all x ∈ ∂ Ω, and that is theonly vector with this property. From the boundary condition (33b), we therefore find D ( a + k g ) = α (90)for some constant α . Hence a = 2 α D − − k g . (91)The periodicity (83) requires a · = 0, so that α = k · g (cid:62) D − = k (cid:101) D, (92)where we have used the property · g = 1 and (86). Applying L k to r k , one finds thenfinds that r k is an eigenfunction with eigenvalue λ ( k ) = 2 (cid:101) D − α ( α + ˜ v ) = k ˜ v + k (cid:101) D. (93)By LF transform, we then obtain the rate function I ( v ) = ( v − ˜ v ) (cid:101) D , (94) arge deviations of currents in diffusions with reflective boundaries v .We further find that the eigenvalue (93) is consistent with assuming for the lefteigenfunction (cid:96) k the form (cid:96) k = exp[ b · x ] , (95)where, in a calculation analogous to the one for r k , we find b = 2 D − ( v + β ) + k g (96)with β = − (cid:62) D − v + k · g (cid:62) D − = − ˜ v − k (cid:101) D. (97)It is interesting to note that the rate function (94) is equivalent to that of a singleparticle with drift ˜ v and diffusivity (cid:101) D . To better understand how current fluctuationsare created, we can determine the effective drift F k of the conditioned dynamics using(23) and the expression for r k . The result is F k = v + ( v − ˜ v ) , (98)which gives for the probability current J F k ,ρ ∗ k ( x ) = v ˜ v J F ,ρ ∗ ( x ) . (99)Thus, for the particle system to generate an atypical fluctuation of the collective current,each particle gives rise through fluctuation to an equal absolute increase in their intrinsicvelocity, equal to ∆ v = v − ˜ v . The current thus changes uniformly.These results are consistent with the fact that the rate function saturates a universalquadratic bound on current fluctuations [41], and are also expected given that theheterogeneous single-file diffusion on a ring, while not satisfying detailed balance directly,does so with respect to a reference frame moving with the collective velocity ˜ v . As aresult, the stationary density of the effective process (20) must be equal to the originalinvariant density (84): ρ ∗ k ( x ) = ρ ∗ ( x ) . (100)This can be checked more directly by noting that a + b = k .To close, it is instructive to compare the results of the single-file diffusion modelto those of the asymmetric simple exclusion process (ASEP) conditioned on a largecurrent [42]. For that model, it was found analytically that asymptotically large currentsare generated through an effective process comprised of two effects: a uniform increase inthe hopping rate of all particles, and a pairwise repulsive interaction between particles,not present in the unconditioned process. Since the ASEP yields in the diffusive limit asingle-file dynamics with identical particles, we conclude that the second effect is purelya lattice effect. Indeed, on the lattice, jammed configurations form a finite fractionof all possible system configuration, whereas on the continuum, jammed configurationsconstitute a boundary layer of measure zero relative to the bulk. arge deviations of currents in diffusions with reflective boundaries
6. Discussion
We have provided two independent methods showing that for reflected diffusions, thecorrect boundary conditions for the spectral problem associated with the dynamical largedeviations of current-like observables are given by (33). These boundary conditions areinteresting in that they mimic the “tilting” of L and L † to L k and L † k , respectively. Weindeed recall that L k is obtained from L by the replacement ∇ → ∇ + k g , while L † k isobtained from L † by the replacement −∇ → −∇ + k g . The same replacements, whenapplied to the condition (27) for f and the condition (26) for ρ , gives, respectively, theboundary condition (33b) for r k and the boundary condition (33a) for (cid:96) k . Based on thisresult, it is natural to conjecture that the same replacements apply to other boundaryconditions describing other types of boundary behavior, e.g., partially reflecting [43] orsticky [44].Two physically significant results follow from the boundary conditions (33). Firstly,they imply that the effective process, describing how current fluctuations are realized,also has reflective boundaries, but relative to the effective drift F k (23). This is expected:all system trajectories are reflected at the boundary and, therefore, so are any subsetof trajectories corresponding to a given current fluctuation. The second, less obviousresult is that the effective force at the boundary is not modified in the normal direction,as expressed in (34). A physical (as opposed to mathematical) understanding of whythis is so may come from studying more specific model diffusions. Note that both resultswere also found for occupation-like observables in one-dimensional diffusions [26], so thetype of observable considered (occupation-like or current-like) is not relevant for theirexplanation.In our example of heterogeneous single-file diffusion conditioned on the collectiveparticle current, the effective force changes in both magnitude and direction, but itsprojection onto the boundary normal does not. One can note that the original processhas an irreversible drift [45], generally defined by F ir ( x ) = J ∗ ( x ) /ρ ∗ ( x ), which isa constant vector, everywhere orthogonal to the boundary normal. This propertyallows us to solve the model exactly [31], and explains why the irreversible drift ofthe effective process is only modified in magnitude [46]. To obtain more complicated—and interesting—behavior upon conditioning on a current, one will need to study modelsfor which the irreversible drift does not have this special structure; looking, for instance,at non-planar boundaries and state-dependent original drifts. Our general results showhow, in principle, one would calculate the large deviation elements in these cases. Inany such model an interesting question will be the relative importance of the systembulk to the near-boundary region in generating the fluctuation. arge deviations of currents in diffusions with reflective boundaries Appendix A. Duality for Markov operators
We show here the calculation leading to the duality relation for the Markov operators: (cid:104) ρ, L f (cid:105) = (cid:10) L † ρ, f (cid:11) − (cid:90) ∂ Ω d x f ( x ) J F ,ρ ( x ) · ˆ n ( x ) − (cid:90) ∂ Ω d x ρ ( x ) D ∇ f ( x ) · ˆ n ( x ) , (A.1)when the process is constrained to a region Ω ⊂ R d . Before proceeding, we state forconvenience a mathematical identity, which amounts to integration by parts in higherdimensions. For a scalar field u and vector field V we have (cid:90) Ω d x u ( x ) ∇ · V ( x ) = − (cid:90) ∂ Ω d x u ( x ) V ( x ) · ˆ n ( x ) − (cid:90) Ω d x V ( x ) · ∇ u ( x ) , (A.2)where care should be taken to note that ˆ n ( x ) is the inward normal vector at point x ∈ ∂ Ω. Starting from the inner product over the domain Ω and using the expression(5) for the Markov generator, we write (cid:104) ρ, L f (cid:105) = (cid:90) Ω d x ρ ( x ) (cid:20) F ( x ) · ∇ + 12 ∇ · D ∇ (cid:21) f ( x ) . (A.3)Using (A.2), we have (cid:90) Ω d x ρ ( x ) F ( x ) · ∇ f ( x ) = − (cid:90) ∂ Ω d x ρ ( x ) f ( x ) F ( x ) · ˆ n ( x ) − (cid:90) Ω d x ∇ · [ F ( x ) ρ ( x )] f ( x )(A.4)and (cid:90) Ω d x ρ ( x ) (cid:20) ∇ · D ∇ (cid:21) f ( x ) = − (cid:90) ∂ Ω d x ρ ( x ) D ∇ f ( x ) · ˆ n ( x ) − (cid:90) Ω d x ∇ ρ ( x ) · D ∇ f ( x ) . (A.5)Given that D is symmetric, we have12 (cid:90) Ω d x ∇ ρ ( x ) · D ∇ f ( x ) = 12 (cid:90) Ω d x D ∇ ρ ( x ) · ∇ f ( x ) (A.6)and applying (A.2) to this last expression, we obtain12 (cid:90) Ω d x D ∇ ρ ( x ) ·∇ f ( x ) = − (cid:90) ∂ Ω d x f ( x ) D ∇ ρ ( x ) · ˆ n ( x ) − (cid:90) Ω d x f ( x ) (cid:20) ∇ · D ∇ (cid:21) ρ ( x ) . (A.7)Substituting (A.4), (A.5) and (A.7) into (A.3), we obtain (cid:104) ρ, L f (cid:105) = (cid:10) L † ρ, f (cid:11) − (cid:90) ∂ Ω d x f ( x ) (cid:20) ρ ( x ) F ( x ) − D ∇ ρ ( x ) (cid:21) · ˆ n ( x ) − (cid:90) ∂ Ω d x ρ ( x ) D ∇ f ( x ) · ˆ n ( x ) , (A.8)where L † is defined as in (3). Recognizing the definition of the current (9) in the above,(A.1) follows. arge deviations of currents in diffusions with reflective boundaries Appendix B. Duality for tilted generators
Here we obtain the duality expression (cid:104) (cid:96) k , L k r k (cid:105) = (cid:68) L † k (cid:96) k , r k (cid:69) − (cid:90) ∂ Ω d x J F k ,(cid:96) k r k ( x ) · ˆ n ( x ) , (B.1)for the tilted generators for a process constrained to a region Ω ⊂ R d , proceeding in asimilar manner as done in Appendix A. Using the expression (16) for the tilted generator,we have (cid:104) (cid:96) k , L k r k (cid:105) = (cid:90) Ω d x (cid:96) k ( x ) (cid:20) F ( x ) · ( ∇ + k g ( x )) + 12 ( ∇ + k g ( x )) · D ( ∇ + k g ( x )) (cid:21) r k ( x ) . (B.2)For the first term we have, using integration by parts as in (A.2), that (cid:90) Ω d x (cid:96) k ( x ) (cid:20) F ( x ) · ( ∇ + k g ( x )) (cid:21) r k ( x ) = − (cid:90) ∂ Ω d x l k ( x ) r k ( x ) F ( x ) · ˆ n ( x )+ (cid:90) Ω d x (cid:20) ( −∇ + k g ( x )) · F ( x ) l k ( x ) (cid:21) r k ( x ) . (B.3)For the second term, we first note that12 ( ∇ + k g ( x )) · D ( ∇ + k g ( x )) = 12 ( ∇ · D ∇ + k g ( x ) · D ∇ + ∇ · k D g ( x ) + k g ( x ) · D g ( x )) . (B.4)The last term in the above contains no derivatives and produces no boundary terms,while the first term has already been dealt with in Appendix A in (A.5) and (A.7) (withthe understanding that (cid:96) k and r k are to replace ρ and f , respectively). We can thereforeimmediately write (cid:90) Ω d x (cid:96) k ( x ) (cid:20) ∇ · D ∇ (cid:21) r k ( x ) = (cid:90) Ω d x r k ( x ) (cid:20) ∇ · D ∇ (cid:21) (cid:96) k ( x ) − (cid:90) ∂ Ω d x (cid:96) k ( x ) D ∇ r k ( x ) · ˆ n ( x )+ 12 (cid:90) ∂ Ω d x r k ( x ) D ∇ (cid:96) k ( x ) · ˆ n ( x ) . (B.5)For the last two terms in (B.4), we have (cid:90) Ω d x (cid:96) k ( x ) (cid:20) k g ( x ) · D ∇ (cid:21) r k ( x ) = (cid:90) Ω d x (cid:96) k ( x ) (cid:20) k D g ( x ) · ∇ (cid:21) r k ( x ) (B.6a)= − (cid:90) ∂ Ω d x (cid:96) k ( x ) r k ( x ) k D g ( x ) · ˆ n ( x ) − (cid:90) Ω d x r k ( x ) ∇ · (cid:96) k ( x ) k D g ( x ) (B.6b) arge deviations of currents in diffusions with reflective boundaries D in the first line and integration by parts in thesecond, and (cid:90) Ω d x (cid:96) k ( x ) (cid:20) ∇ · k D g ( x ) (cid:21) r k ( x ) = − (cid:90) ∂ Ω d x (cid:96) k ( x ) r k ( x ) k D g ( x ) − (cid:90) Ω d x (cid:20) k D g ( x ) · ∇ (cid:96) k ( x ) (cid:21) r k ( x ) . (B.7)Combining (B.3), (B.5), (B.6) and (B.7) we obtain (cid:104) (cid:96) k , L k r k (cid:105) = (cid:90) Ω d x r k ( x ) (cid:20) ( −∇ + k g ( x )) · ( F ( x ) (cid:96) k ( x )) + 12 (cid:18) ∇ · D ∇ (cid:96) k ( x ) − k D g ( x ) · ∇ (cid:96) k ( x ) − ∇ · ( k D g ( x ) (cid:96) k ( x )) + k g ( x ) · D g ( x ) (cid:96) k ( x ) (cid:19)(cid:21) − (cid:90) ∂ Ω d x (cid:26) (cid:96) k ( x ) r k ( x ) (cid:18) F ( x ) + k D g ( x ) (cid:19) + 12 (cid:96) k ( x ) D ∇ r k ( x ) − r k ( x ) D ∇ (cid:96) k ( x ) (cid:27) · ˆ n ( x ) . (B.8)Noting from (22) that the expression in square brackets is simply the operator L † k appliedto (cid:96) k , and observing that (cid:96) k r k (cid:18) F + k D g (cid:19) + 12 (cid:96) k D ∇ r k − r k D ∇ (cid:96) k = J F k ,(cid:96) k r k (B.9)where we have used the definition of the current (9) and effective force (23), we obtainthe duality relation (B.1). Acknowledgments
Emil Mallmin acknowledges studentship funding from EPSRC Grant No. EP/N509644/1.
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