[Paper Review] Exact Large Deviation Functional of a Stationary Open Driven Diffusive System: The Asymmetric Exclusion Process

Bernard Derrida, Joel L. Lebowitz+1 more|arXiv (Cornell University)|May 16, 2002

Stochastic processes and statistical mechanics14 references18 citations

TL;DR

This paper derives the exact large deviation functional (LDF) for the asymmetric exclusion process (ASEP) in a stationary open system with particle reservoirs at both ends. Using exact microscopic dynamics and a variational principle, it shows that the LDF is non-local and non-convex in certain parameter regimes, leading to non-Gaussian fluctuations and phase transitions, generalizing earlier results for the symmetric case to driven, nonequilibrium systems with shocks and discontinuities in second derivatives of the LDF.

ABSTRACT

We consider the asymmetric exclusion process (ASEP) in one dimension on sites $i = 1,..., N$, in contact at sites $i=1$ and $i=N$ with infinite particle reservoirs at densities $ρ_a$ and $ρ_b$. As $ρ_a$ and $ρ_b$ are varied, the typical macroscopic steady state density profile $\bar ρ(x)$, $x\in[a,b]$, obtained in the limit $N=L(b-a) o\infty$, exhibits shocks and phase transitions. Here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile $ρ(x)$: $P_N(\{ρ(x)\})\sim\exp[-L{\cal F}_{[a,b]}(\{ρ(x)\});ρ_a,ρ_b]$, so that ${\cal F}$ is the large deviation functional, a quantity similar to the free energy of equilibrium systems. We find, as in the symmetric, purely diffusive case $q=1$ (treated in an earlier work), that $\cal F$ is in general a non-local functional of $ρ(x)$. Unlike the symmetric case, however, the asymmetric case exhibits ranges of the parameters for which ${\cal F}(\{ρ(x)\})$ is not convex and others for which ${\cal F}(\{ρ(x)\})$ has discontinuities in its second derivatives at $ρ(x) = \barρ(x)$; the fluctuations near $\barρ(x)$ are then non-Gaussian and cannot be calculated from the large deviation function.

Motivation & Objective

• To derive the exact large deviation functional (LDF) for the asymmetric exclusion process (ASEP) in a stationary open system with particle reservoirs at both ends.
• To understand the statistical mechanics of large deviations in nonequilibrium steady states (SNS), particularly the emergence of non-Gaussian fluctuations and phase transitions.
• To generalize previous results on symmetric exclusion processes to the asymmetric case, revealing new features such as non-convexity and discontinuities in the second derivative of the LDF.

Proposed method

• The authors use the exact microscopic transition rates of the ASEP to compute the probability of observing a macroscopic density profile ρ(x) in the steady state.
• They derive the large deviation functional F[a,b]({ρ(x)}; ρa, ρb) as the limit lim_{L→∞} log(PN)/L, where PN is the probability of observing profile ρ(x) over L(b−a) sites.
• A variational principle is applied to compute the supremum over auxiliary functions F(x), leading to the construction of an optimizing function Fρ(x) via the concave envelope of an integral of (1−ρ(y))dy.
• The functional F is expressed as a sum of a bulk entropy term and a non-local correction involving the concave envelope Gρ and the optimized Fρ, with constraints on boundary densities ρa and ρb.
• The method relies on the matrix product ansatz and exact solution techniques for the ASEP, extended to compute large deviation probabilities beyond typical behavior.
• The analysis includes rigorous justification of the concave envelope construction and the optimality of Fρ(x) through integration by parts and convexity arguments.

Experimental results

Research questions

• RQ1 How does the large deviation functional (LDF) for the asymmetric exclusion process (ASEP) differ from that of the symmetric case in terms of convexity and non-locality?
• RQ2 What are the conditions under which the LDF exhibits non-Gaussian fluctuations due to discontinuities in its second derivative?
• RQ3 How do shocks and phase transitions in the typical density profile affect the structure of the LDF in open, driven diffusive systems?
• RQ4 Can the LDF for the ASEP be derived exactly from microscopic dynamics, and how does it depend on reservoir densities ρa and ρb?
• RQ5 What role does the concave envelope of the integrated density profile play in determining the optimal fluctuation profile?

Key findings

• The large deviation functional F for the ASEP is non-local and non-convex in certain parameter regimes, indicating non-Gaussian fluctuations that cannot be captured by a Gaussian approximation.
• The LDF exhibits discontinuities in its second derivative at ρ(x) = ρ̄(x), leading to non-Gaussian behavior in the fluctuations around the typical profile.
• For ρa ≠ ρb, the LDF is derived exactly as F[a,b]({ρ(x)}; ρa, ρb) = −(b−a)K(ρa,ρb) + ∫[ρ(x)logρ(x) + (1−ρ(x))log(1−ρ(x))]dx + sup_F Bρ(F), with Bρ(F) involving an optimized function Fρ(x).
• The optimal profile ρ(x) minimizing the LDF under a fixed mean density constraint is monotone non-decreasing when ρa > ρb, and piecewise constant when ρa < ρb, with a phase transition at critical densities.
• The functional F is pointwise convex in ρ(x) when ρ(x) = 1 − G′_ρ(x), and the minimum is achieved only when ρ(x) is constant if the profile is to minimize the LDF under a fixed mean density constraint.
• The derivation confirms that the LDF for the ASEP generalizes the equilibrium free energy form but includes non-local corrections due to the driven, nonequilibrium nature of the system.