33 research outputs found
Bethe Ansatz and Q-operator for the open ASEP
In this paper, we look at the asymmetric simple exclusion process with open
boundaries with a current-counting deformation. We construct a two-parameter
family of transfer matrices which commute with the deformed Markov matrix of
the system. We show that these transfer matrices can be factorised into two
commuting matrices with one parameter each, which can be identified with
Baxter's Q-operator, and that for certain values of the product of those
parameters, they decompose into a sum of two commuting matrices, one of which
is the Bethe transfer matrix for a given dimension of the auxiliary space.
Using this, we find the T-Q equation for the open ASEP, and, through functional
Bethe Ansatz techniques, we obtain an exact expression for the dominant
eigenvalue of the deformed Markov matrix.Comment: 46 pages. New version: references updated and typos correcte
Deriving GENERIC from a generalized fluctuation symmetry
Much of the structure of macroscopic evolution equations for relaxation to
equilibrium can be derived from symmetries in the dynamical fluctuations around
the most typical trajectory. For example, detailed balance as expressed in
terms of the Lagrangian for the path-space action leads to gradient zero-cost
flow. We find a new such fluctuation symmetry that implies GENERIC, an
extension of gradient flow where a Hamiltonian part is added to the dissipative
term in such a way as to retain the free energy as Lyapunov function
Frenetic aspects of second order response
Starting from second order around thermal equilibrium, the response of a
statistical mechanical system to an external stimulus is not only governed by
dissipation and depends explicitly on dynamical details of the system. The so
called frenetic contribution in second order around equilibrium is illustrated
in different physical examples, such as for non-thermodynamic aspects in the
coupling between system and reservoir, for the dependence on disorder in
dielectric response and for the nonlinear correction to the
Sutherland--Einstein relation. More generally, the way in which a system's
dynamical activity changes by the pertubation is visible (only) from nonlinear
response.Comment: 32 pages, 6 figure
Exact Current Statistics of the ASEP with Open Boundaries
Non-equilibrium systems are often characterized by the transport of some
quantity at a macroscopic scale, such as, for instance, a current of particles
through a wire. The Asymmetric Simple Exclusion Process (ASEP) is a paradigm
for non-equilibrium transport that is amenable to exact analytical solution. In
the present work, we determine the full statistics of the current in the finite
size open ASEP for all values of the parameters. Our exact analytical results
are checked against numerical calculations using DMRG techniques.Comment: 5 pages, references adde
Large deviations and dynamical phase transitions in stochastic chemical networks
Chemical reaction networks offer a natural nonlinear generalisation of linear
Markov jump processes on a finite state-space. In this paper, we analyse the
dynamical large deviations of such models, starting from their microscopic
version, the chemical master equation. By taking a large-volume limit, we show
that those systems can be described by a path integral formalism over a
Lagrangian functional of concentrations and chemical fluxes. This Lagrangian is
dual to a Hamiltonian, whose trajectories correspond to the most likely
evolution of the system given its boundary conditions. The same can be done for
a system biased on time-averaged concentrations and currents, yielding a biased
Hamiltonian whose trajectories are optimal paths conditioned on those
observables. The appropriate boundary conditions turn out to be mixed, so that,
in the long time limit, those trajectories converge to well-defined attractors.
We are then able to identify the largest value that the Hamiltonian takes over
those attractors with the scaled cumulant generating function of our
observables, providing a non-linear equivalent to the well-known
Donsker-Varadhan formula for jump processes. On that basis, we prove that
chemical reaction networks that are deterministically multistable generically
undergo first-order dynamical phase transitions in the vicinity of zero bias.
We illustrate that fact through a simple bistable model called the Schl\"ogl
model, as well as multistable and unstable generalisations of it, and we make a
few surprising observations regarding the stability of deterministic fixed
points, and the breaking of ergodicity in the large-volume limit