Abstract:
We study spacetime thermodynamics for non-equilibrium processes. We first generalize the formulation of spacetime thermodynamics by using an observer outside the horizon. Then we construct the entropy balance equation of spacetime thermodynamics for non-equilibrium processes in f(R) gravity. The coefficients of the expansion and shear terms are equal to the viscosities of the black hole membrane paradigm, and a new entropy production term appears.

Abstract:
Basic concepts like energy, heat, and temperature have acquired a precise meaning after the development of thermodynamics. Thermodynamics provides the basis for understanding how heat and work are related and with the general rules that the macroscopic properties of systems at equilibrium follow. Outside equilibrium and away from macroscopic regimes most of those rules cannot be applied directly. In this paper we present recent developments that extend the applicability of thermodynamic concepts deep into mesoscopic and irreversible regimes. We show how the probabilistic interpretation of thermodynamics together with probability conservation laws can be used to obtain kinetic equations describing the evolution of the relevant degrees of freedom. This approach provides a systematic method to obtain the stochastic dynamics of a system directly from the knowledge of its equilibrium properties. A wide variety of situations can be studied in this way, including many that were thought to be out of reach of thermodynamic theories, such as non-linear transport in the presence of potential barriers, activated processes, slow relaxation phenomena, and basic processes in biomolecules, like translocation and stretching.

Abstract:
Local Shannon entropy lies at the heart of modern thermodynamics, with much discussion of trajectory-dependent entropy production. When taken at both boundaries of a process in phase space, it reproduces the second law of thermodynamics over a finite time interval for small scale systems. However, given that entropy is an ensemble property, it has never been clear how one can assign such a quantity locally. Given such a fundamental omission in our knowledge, we construct a new ensemble composed of trajectories reaching an individual microstate, and show that locally defined entropy, information, and free energy are properties of the ensemble, or trajectory-independent true thermodynamic potentials. We find that the Boltzmann-Gibbs distribution and Landauer's principle can be generalized naturally as properties of the ensemble, and that trajectory-free state functions of the ensemble govern the exact mechanism of non-equilibrium relaxation.

Abstract:
We discuss inertial effects in systems outside equilibrium within the framework of non-equilibrium thermodynamics. By introducing a Gibbs equation in which the entropy depends on the probability density, we are able to describe a system of Brownian particles immersed in a heat bath in both inertial and diffusion regimes. In the former, a relaxation equation for the diffusion current is obtained whereas in the latter we recover Fick's law. Our approach, which uses the elements of the theory of internal degrees of freedom, constitutes the mesoscopic version of a previous analysis which takes into account the kinetic energy of diffusion.

Abstract:
It is known that equilibrium thermodynamics can be deduced from a constrained Fisher information extemizing process. We show here that, more generally, both non-equilibrium and equilibrium thermodynamics can be obtained from such a Fisher treatment. Equilibrium thermodynamics corresponds to the ground state solution, and non-equilibrium thermodynamics corresponds to excited state solutions, of a Schroedinger wave equation (SWE). That equation appears as an output of the constrained variational process that extremizes Fisher information. Both equilibrium- and non-equilibrium situations can thereby be tackled by one formalism that clearly exhibits the fact that thermodynamics and quantum mechanics can both be expressed in terms of a formal SWE, out of a common informational basis.

Abstract:
What is really measured in dynamic calorimetric experiments is still an open question. This paper is devoted to this question, which can be usefully envisaged by means of macroscopic non-equilibrium thermodynamics. From the pioneer work of De Donder on chemical reactions and with other authors along the 20th century, the question is tackled under an historical point of view. A special attention is paid about the notions of frequency dependent complex heat capacity and entropy production due to irreversible processes occurring during an experiment. This phenomenological approach based on thermodynamics, not widely spread in the literature of calorimetry, could open significant perspectives on the study of macro-systems undergoing physico-chemical transformations probed by dynamic calorimetry.

Abstract:
In the last ten years, a number of ``Conventional Fluctuation Theorems'' have been derived for systems with deterministic or stochastic dynamics, in a transient or in a non-equilibrium stationary state. These theorems gave explicit expressions for the ratio of the probability to find the system with a certain value of entropy (or heat) production to that of finding the opposite value. A similar theorem for the fluctuations of the work done on a system has recently been demonstrated experimentally for a simple system in a transient state, consisting of a Brownian particle in water, confined by a moving harmonic potential. In this paper we show that because of the interaction between the stochastic motion of the particle in water and its deterministic motion in the potential, very different new heat theorems are found than in the conventional case. One of the consequences of these new heat Fluctuation Theorems is that the ratio of the probability for the Brownian particle to absorb heat from rather than supply heat to the water is much larger than in the Conventional Fluctuation Theorem. This could be of relevance for micro/nano-technology.

Abstract:
The second entropy theory for non-equilibrium thermodynamics is used to show that the optimum structure or pattern of a time-dependent system corresponds to the maximum entropy. A formula for the total entropy of convective heat flow is derived. The Navier-Stokes equations in Boussinesq approximation for straight roll convection are solved by a Fourier expansion technique. Results for the velocity amplitude are in good agreement with previous computations and experimental measurements. For the spontaneous transitions between straight roll states reported in the literature, it is shown that the measured change in convective pattern wave length corresponds to an increase in the entropy.

Abstract:
We briefly review the concept of non-equilibrium temperature from the perspectives of extended irreversible thermodynamics, fluctuation theory, and statistical mechanics. The relations between different proposals are explicitly examined in two especially simple systems: an ideal gas in steady shear flow and a forced harmonic oscillator in a thermal bath. We examine with special detail temperatures related to the average molecular kinetic energy along different spatial directions, to the average configurational energy, to the derivative of the entropy with respect to internal energy, to fluctuation-dissipation relation and discuss their measurement.

Abstract:
The second entropy theory for non-equilibrium thermodynamics is used to show that the optimum structure or pattern of a time-dependent system corresponds to the maximum entropy. A formula for the total entropy of convective heat flow is derived. The Navier-Stokes equations in Boussinesq approximation for straight roll convection are solved by a Fourier expansion technique. Results for the velocity amplitude are in good agreement with previous computations and experimental measurements. For the spontaneous transitions between straight roll states reported in the literature, it is shown that the measured change in convective pattern wave length corresponds to an increase in the entropy. This paper has been superseded by arXiv:1208.5105v1.