We study the critical behavior of the entropy production of the Ising model subject to a magnetic field that oscillates in time. The mean-field model displays a phase transition that can be either first or second-order, depending on the amplitude of the field and on the frequency of oscillation. Within this approximation the entropy production rate is shown to have a discontinuity when the transition is first-order and to be continuous, with a jump in its first derivative, if the transition is second-order. In two dimensions, we find with numerical simulations that the critical behavior of the entropy production rate is the same, independent of the frequency and amplitude of the field. Its first derivative has a logarithmic divergence at the critical point. This result is in agreement with the lack of a first-order phase transition in two dimensions. We analyze a model with a field that changes at stochastic time-intervals between two values. This model allows for an informational theoretic interpretation, with the system as a sensor that follows the external field. We calculate numerically a lower bound on the learning rate, which quantifies how much information the system obtains about the field. Its first derivative with respect to temperature is found to have a jump at the critical point.

Systems kept out of equilibrium in stationary states by an external source of energy store an energy ∆U = U − U 0 . U 0 is the internal energy at equilibrium state, obtained after the shutdown of energy input.We determine ∆U for two model systems: ideal gas and Lennard-Jones fluid. ∆U depends not only on the total energy flux, J U , but also on the mode of energy transfer into the system. We use three different modes of energy transfer where: the energy flux per unit volume is (i) constant; (ii) proportional to the local temperature (iii) proportional to the local density. We show that ∆U/J U = τ is minimized in the stationary states formed in these systems, irrespective of the mode of energy transfer. τ is the characteristic time scale of energy outflow from the system immediately after the shutdown of energy flux. We prove that τ is minimized in stable states of the Rayleigh-Benard cell.Systems out of equilibrium are notoriously difficult to describe in a single coherent methodology based on variational principles. Principles such as Prigogine minimum entropy production [1], Attard second entropy variation [2] or, Ziegler maximum entropy production [3] etc. suggested over the last 100 years, have not reached the same status as the maximum entropy principle known from equilibrium thermodynamics [4][5][6]. A new paradigm, such as the driven lattice gas system, is believed to become an "Ising model" for non-equilibrium statistical physics [7][8][9][10][11][12]. Steady State Thermodynamics (SST) is yet another description framework for non-equilibrium stationary states, which is still being developed [13][14][15][16]. Here we present a different approach to stationary states, based on two quantities: the energy stored in non-equilibriums states, ∆U, and the total energy flux, J U in these states.The second law of thermodynamics states that the entropy of a system has its maximum value at the equilibrium state. Entropy, S , is a function of state, thus for an isolated system of N molecules of total internal energy U enclosed in a volume, V , the entropy has a fixed value S = S (U, V, N).

We discovered an out-of-equilibrium transition in the ideal gas between two walls, divided by an inner, adiabatic, movable wall. The system is driven out-of-equilibrium by supplying energy directly into the volume of the gas. At critical heat flux, we have found a continuous transition to the state with a low-density, hot gas on one side of the movable wall and a dense, cold gas on the other side. Molecular dynamic simulations of the soft-sphere fluid confirm the existence of the transition in the interacting system. We introduce a stationary state Helmholtz-like function whose minimum determines the stable positions of the internal wall. This transition can be used as a paradigm of transitions in stationary states and the Helmholtz-like function as a paradigm of the thermodynamic description of these states.