From Clausius's principle, the entropy change in a closed system satisfies ΔS ⩾ ∑kQk/Tk, where the equality holds for reversible processes and the inequality holds for irreversible processes. However, the entropy changes of infinite heat reservoirs are routinely calculated as ΔS = Q/T, independently of the nature of the heat exchange, whether reversible or irreversible. We show how this result comes about by considering the direct thermal contact between two systems, and then by allowing one of them to be much larger than the other. We then re-examine the usual definition of reversibility to reconcile it with the expression found for the entropy change.

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We report on exact results for stochastic Ising systems that evolve in time due to a simultaneous action of several independent spin-flip mechanisms. The transition probability describes a competition between different values for the two parameters characterizing the involved (nearest-neighbour) Hamiltonian, namely, the interaction strength and the applied magnetic field. Such a conflicting dynamics aims to represent the sort of diffusing microscopic disorder that may occur in real systems. We determine the steady states for a class of transition probabilities for one-dimensional lattices; they exhibit amazing critical behaviour. Furthermore, it is shown that the steady state for a singular class of transition probabilities may be represented by an effective Hamiltonian that has the Ising structure with complex parameters for any lattice dimension.Let U { s, = rt 1; r E Z d } denote any spin (equivalently, particle, neuron, etc.) configura-(1) e.g., due to the coupling of U to a heat bath at temperature T. This induces a sequence of flips, i.e. U ' represents U after flipping the spin at site r: s, +s, . Glauber's kinetic version of the Ising model [3] occurs if the probability per unit time or rate for the transitions from U to or is c ( d lo) = C(U' IQ; <), where the latter satisfies detailed balance, namely, ~( a ' 15; C) = = ~( c l d ; C)exp[ -PAH:]. Here, / 3 = (kB T)-', kB is Boltzmann's constant, AH: = H(a'; <) --H(o; C), and

We have developed a simple, systematic method to investigate the existence of stationary probability distributions (SPDS) for interacting particle (or spin) lattice systems exhibiting steady non-equilibrium states. The latter originate in a competition between several creation-annihilation (spin-flip) kinetic mechanisms, say each acting presuming a different bath temperature, particle (spin) interactions strength or chemical potential (magnetic field). It follows the existence of SPDS for a large class of these systems which may thus be studied by simply applying the techniques of equilibrium theory. The method is illustrated with several examples bearing a practical interest.

We show that the claim that our definition of reversibility for processes in a system is incorrect is based on erroneous or irrelevant reasoning.