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We present the closed loop approach to linear nonequilibrium thermodynamics considering a generic heat engine dissipatively connected to two temperature baths. The system is usually quite generally characterized by two parameters: the output power $P$ and the conversion efficiency $eta$, to which we add a third one, the working frequency $omega$. We establish that a detailed understanding of the effects of the dissipative coupling on the energy conversion process, necessitates the knowledge of only two quantities: the systems feedback factor $beta$ and its open-loop gain $A_{0}$, the product of which, $A_{0}beta$, characterizes the interplay between the efficiency, the output power and the operating rate of the system. By placing thermodynamics analysis on a higher level of abstraction, the feedback loop approach provides a versatile and economical, hence a very efficient, tool for the study of emph{any} conversion engine operation for which a feedback factor may be defined.
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We develop the stochastic approach to thermodynamics based on the stochastic dynamics, which can be discrete (master equation) continuous (Fokker-Planck equation), and on two assumptions concerning entropy. The first is the definition of entropy itself and the second, the definition of entropy production rate which is nonnegative and vanishes in thermodynamic equilibrium. Based on these assumptions we study interacting systems with many degrees of freedom in equilibrium or out of thermodynamic equilibrium, and how the macroscopic laws are derived from the stochastic dynamics. These studies include the quasi-equilibrium processes, the convexity of the equilibrium surface, the monotonic time behavior of thermodynamic potentials, including entropy, the bilinear form of the entropy production rate, the Onsager coefficients and reciprocal relations, and the nonequilibrium steady states of chemical reactions.
The standard formulation of thermostatistics, being based on the Boltzmann-Gibbs distribution and logarithmic Shannon entropy, describes idealized uncorrelated systems with extensive energies and short-range interactions. In this letter, we use the fundamental principles of ergodicity (via Liouvilles theorem), the self-similarity of correlations, and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long-range-interacting systems. Significantly, our formalism provides a justification for the well-studied nonextensive thermostatistics characterized by the Tsallis distribution, which it includes as a special case. We also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the Boltzmann-Gibbs-Shannon entropy. The consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics.
A theoretical framework is developed for the phenomenon of non-Gaussian normal diffusion that has experimentally been observed in several heterogeneous systems. From the Fokker-Planck equation with the dynamical structure with largely separated time scales, a set of three equations are derived for the fast degree of freedom, the slow degree of freedom and the coupling between these two hierarchies. It is shown that this approach consistently describes diffusing diffusivity and non-Gaussian normal diffusion.
Non-equilibrium processes in Schottky systems generate by projection onto the equilibrium subspace reversible accompanying processes for which the non-equilibrium variables are functions of the equilibrium ones. The embedding theorem which guarantees the compatibility of the accompanying processes with the non-equilibrium entropy is proved. The non-equilibrium entropy is defined as a state function on the non-equilibrium state space containing the contact temperature as a non-equilibrium variable. If the entropy production does not depend on the internal energy, the contact temperature changes into the thermostatic temperature also in non-equilibrium, a fact which allows to use temperature as a primitive concept in non-equilibrium. The dissipation inequality is revisited, and an efficiency of generalized cyclic processes beyond the Carnot process is achieved.
State functions play important roles in thermodynamics. Different from the process function, such as the exchanged heat $delta Q$ and the applied work $delta W$, the change of the state function can be expressed as an exact differential. We prove here that, for a generic thermodynamic system, only the inverse of the temperature, namely $1/T$, can serve as the integration factor for the exchanged heat $delta Q$. The uniqueness of the integration factor invalidates any attempt to define other state functions associated with the exchanged heat, and in turn, reveals the incorrectness of defining the entransy $E_{vh}=C_VT^2 /2$ as a state function by treating $T$ as an integration factor. We further show the errors in the derivation of entransy by treating the heat capacity $C_V$ as a temperature-independent constant.
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