A major limitations for many heat engines is that their functioning demands on-line control, and/or an external fitting between environmental parameters (e.g. temperatures of thermal baths) and internal parameters of the engine. We study a model for an adaptive heat engine, where-due to feedback from the functional part-the engine's structure adapts to given thermal baths. Hence no on-line control and no external fitting are needed. The engine can employ unknown resources; it can also adapt to results of its own functioning that makes the bath temperatures closer. We determine thermodynamic costs of adaptation and relate them to the prior information available about the environment. We also discuss informational constraints on the structure-function interaction that are necessary for adaptation.

PACS 05.20.-y -statistical mechanics PACS 05.10.Gg -stochastic models in statistical mechanics and non-linear dynamics PACS 05.70.Ln -irreversible thermodynamics Abstract -We study a class of non-equilibrium quasi-stationary states for a Markov system interacting with two different thermal baths. We show that the work done under a slow, external change of parameters admits a potential, i.e., the free energy. Three conditions are needed for the existence of free energy in this non-equilibrium system: time-scale separation between variables of the system, partial controllability (external fields couple only with the slow variable), and an effective detailed balance. These conditions are facilitated in the continuous limit for the slow variable. In contrast to its equilibrium counterpart, the non-equilibrium free energy can increase with temperature. One example of this is that entropy reduction by means of external fields (cooling) can be easier (in the sense of the work cost) if it starts from a higher temperature.

An important statistical test on the pseudo-random number generators is called the spectral test. The test is aimed at answering the question of distribution of the generated pseudo-random vectors in dimensions d that are larger than the genuine dimension of a generator N. In particular, the default MIXMAX generators have various dimensions: N = 8, 17, 240 and higher. Therefore the spectral test is important to perform in dimensions d > 8 for N = 8 generator, d > 17 for N = 17 and d > 240 for N = 240 generator. These tests have been performed by L'Ecuyer and collaborators. When d > N the vectors of the generated numbers fall into the parallel hyperplanes and the distances between them can be larger than the genuine resolving power of the MIXMAX generators, which is l = 2 −61 . The aim of this article is to further study the spectral properties of the MIXMAX generators, to investigate the dependence of the spectral properties of the MIXMAX generators as a function of their internal parameters and in particular their dependence on the parameter m. We found that the best spectral properties are realised when m is between 2 24 and 2 36 , a range which is inclusive of the value of the N = 17 generator. We also provide the alternative parameters for the generators, N = 8 and N = 240 with m in this optimised range.1 On a leave of absence from the A.I.

The no-pumping theorem refers to a Markov system that holds the detailed balance, but is subject to a time-periodic external field. It states that the time-averaged probability currents nullify in the steady periodic (Floquet) state, provided that the Markov system holds the Arrhenius transition rates. This makes an analogy between features of steady periodic and equilibrium states, because in the latter situation all probability currents vanish explicitly. However, the assumption on the Arrhenius rates is fairly specific, and it need not be met in applications. Here a new mechanism is identified for the no-pumping theorem, which holds for symmetric time-periodic external fields and the so called destination rates. These rates are the ones that lead to the locally equilibrium form of the master equation, where dissipative effects are proportional to the difference between the actual probability and the equilibrium (Gibbsian) one. The mechanism also leads to an approximate no-pumping theorem for the Fokker-Planck rates that relate to the discrete-space Fokker-Planck equation.