Cusp Singularities in Boundary-Driven Diffusive Systems
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Cusp Singularities in Boundary-Driven Diffusive Systems Guy Bunin · Yariv Kafri · Daniel Podolsky
Received: 17 January 2013 / Accepted: 5 April 2013 / Published online: 7 May 2013 © Springer Science+Business Media New York 2013
Abstract Boundary driven diffusive systems describe a broad range of transport phenomena. We study large deviations of the density profile in these systems, using numerical and analytical methods. We find that the large deviation may be non-differentiable, a phenomenon that is unique to non-equilibrium systems, and discuss the types of models which display such singularities. The structure of these singularities is found to generically be a cusp, which can be described by a Landau free energy or, equivalently, by catastrophe theory. Connections with analogous results in systems with finite-dimensional phase spaces are drawn. Keywords Boundary-driven diffusive systems · Rare events · Large deviations · Phase-transitions · Catastrophe theory
1 Introduction and Framework The dynamics in many systems of physical interest are described by a field ρ(x, t), with large-scale conserving diffusive behavior and noise. For example, ρ(x, t) could describe the density of diffusing particles, the local temperature in a heat transport experiment, or any other field which behaves diffusively. For such systems, when the interactions are short range, it is accepted [3, 11, 26, 39, 40], that the large-scale behavior of the current obeys Fick’s- (or Ohm’s- or Fourier’s-) law with noise. Here our interest is in transport experiments, where the system is attached to reservoirs, whose effect is to fix the value of ρ at the boundaries, resulting in a net current flowing down the gradient. For such systems the density ρ(x, t) and the current J (x, t) satisfy the conservation relation ∂t ρ + ∂x J = 0, G. Bunin () · Y. Kafri · D. Podolsky Technion—Israel Institute of Technology, Haifa 32000, Israel e-mail: [email protected]
(1)
Cusp Singularities in Boundary-Driven Diffusive Systems
where
J = −D ρ(x, t) ∂x ρ(x, t) + σ ρ(x, t) η(x, t).
113
(2)
Here D(ρ(x, t)) is a density-dependent diffusivity function, and σ (ρ(x, t)) controls the amplitude of the white noise η(x, t), which satisfies η(x, t) = 0 and η(x, t)η(x , t ) = N −1 δ(x − x )δ(t − t ). Here we restrict ourselves to temperatures above any phase-transitions (e.g., transitions to ordered phases such as crystals). In this case [39], D(ρ) and σ (ρ) are smooth functions, and D > 0. For simplicity we consider one dimension, where the distance is rescaled by the system size N , so that 0 ≤ x ≤ 1, and time is rescaled by N 2 . The small N −1 term in the noise is a direct consequence of this coarse-graining. D(ρ) and σ (ρ) are related via a fluctuation-dissipation relation, which for particle systems reads σ (ρ) = 2kB T ρ 2 κ(ρ)D(ρ) where κ(ρ) is the compressibility [11]. The system is attached to reservoirs at the boundaries x = 0, 1, which act as boundary conditions (BCs), ρ(x = 0, t) = ρL and ρ(x = 1, t) = ρR . If ρL = ρR a curre
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