ASEP on a ring conditioned on enhanced flux

Abstract

We show that in the asymmetric simple exclusion process (ASEP) on a ring, conditioned on carrying a large flux, the particle experience an effective longrange potential which in the limit of very large flux takes the simple form U = −2 i =j log | sin π(n i /L − n j /L)|, where n 1 , n 2 , . . . n N are the particle positions, similar to the effective potential between the eigenvalues of the circular unitary ensemble in random matrices. Effective hopping rates and various quasistationary probabilities under such a conditioning are found analytically using the Bethe ansatz and determinantal free fermion techniques. Our asymptotic results extend to the limit of large current and large activity for a family of reaction-diffusion processes with onsite exclusion between particles. We point out an intriguing generic relation between classical stationary probability distributions for conditioned dynamics and quantum ground state wave functions, in particular, in the case of exclusion processes, for free fermions.

Large deviations of the current for driven periodic diffusions

“…is not a priori satisfied, since J T is only proportional to Σ T in the limit T → ∞ because of the potential boundary terms in (29). For V 0 = 0, however, J T is exactly proportional to Σ T for all T , so the operator symmetry (38) holds, as can be verified from the expression (14) of L k .…”

“…which implies, by applying the change of variables (29) and by neglecting the potential boundary terms [62],…”

“…This process is physically important as it describes, by means of a modified stochastic process, how fluctuations of the current or any time-integrated observable in general are created in time [23][24][25][26][27][28]. This effective description of fluctuations was illustrated recently in the context of interacting particle systems [29][30][31][32][33], diffusions [34][35][36], and quantum systems [37][38][39][40][41]. For the SDE (1), preliminary results [26] have shown that the auxiliary process modifies not only the driving γ, which is a natural way to increase or decrease the current, but also the potential V (θ) in a non-local and nonlinear way.…”

Exact Matrix Product Solution for the Boundary-Driven LindbladXXZChain

“…In this way one readily obtains σ x n = σ y n = j x = j y = 0. In the isotropic model with a twist the parity symmetry is broken, but its place is taken by another symmetry, which we specify here for twisting angles θ R = −θ L = π/2: It involves left-right reflection R(A⊗B ⊗...⊗C) = (C ⊗....⊗B ⊗A)R, global rotation in the XY -plane U rot = diag(1, i) ⊗N and Σ x = (σ x ) ⊗N and reads ρ = V ρV † , where V = Σ x U rot R [20]. It is straightforward to check that neither of the set of observables σ α n , j α , changes sign under the V symmetry, and therefore they are generically nonzero.…”

Symmetries and Geometrical Properties of Dynamical Fluctuations in Molecular Dynamics

“…We imagine that the noise force db in (19) comes from friction between the particle and a surrounding solvent, but now imagine that the solvent is moving with constant velocity v. We refer to this as a system with advection (of the particle, by the solvent). In this case the equations of motion are obtained by applying a Galilean transformation to (19), which yields dq t = p t dt, dp t = db…”