Large deviations of the entropy production rate for a class of Gaussian processes

Abstract

We prove a large deviation principle (LDP) and a fluctuation theorem for the entropy production rate (EPR) of the following d dimensional stochastic differential equation dXt=AXtdt+QdBt, where A is a real normal stable matrix, Q is positive definite, and the matrices A and Q commute. The rate function for the EPR takes the following explicit form: I(x)=x1+ℓ0(x)−12+12∑k=1dαk2−βk2ℓ0(x)+αk for x ≥ 0 and I(x)=−x1+ℓ0(x)+12+12∑k=1dαk2−βk2ℓ0(x)+αk for x < 0, where αk ±iβk are the eigenvalues of A and ℓ0(x) is the unique solution of the equation x=1+ℓ×∑k=1dβk2αk2−ℓβk2,−1≤ℓ<mink=1,…,dαk2βk2. Simple closed form formulas for rate functions are rare, and our work identifies an important class of large deviation problems where such formulas are available. The logarithmic moment generating function (the fluctuation function) Λ associated with the LDP is given as Λ(λ)=−12∑k=1dαk2−4λ(1+λ)βk2+αk for λ∈D and Λ(λ) = ∞ for λ∉D, where D is the domain of Λ. The functions Λ(λ) and I(x) satisfy the Cohen–Gallavotti symmetry properties: Λ(x)=Λ(−(1+x)),I(x)=I(−x)−x, for all x∈R. In particular, the functions I and Λ do not depend on the diffusion matrix Q and are determined completely by the real and imaginary parts of the eigenvalues of A. Formally, the deterministic system with Q = 0 has zero EPR, and thus, the model exhibits a phase transition in that the EPR changes discontinuously at Q = 0.