Massieu function

Abstract: The thermodynamics of three-dimensional asymptotically flat cosmological solutions that play the same role than the BTZ black holes in the anti-de Sitter case is derived and explained from holographic properties of flat space. It is shown to coincide with the flat-space limit of the thermodynamics of the inner black hole horizon on the one hand and the semi-classical approximation to the gravitational partition function associated to the entropy of the outer horizon on the other. This leads to the insight that it is the Massieu function that is universal in the sense that it can be computed at either horizon.

Abstract: The axiomatic thermodynamics of Giles (1964) is extended in such a way that direct applications to open systems are possible. The Helmholtz and other generalised Massieu function representations of thermodynamics are discussed without specialising to equilibrium. Quasi, 'ordinary', and absolute Helmholtz and generalised Massieu functions are introduced and shown to exist. 'Uniqueness' theorems for these quantities are established. A fundamental theorem is proved for each of the above representations of thermodynamics. This theorem provides quantitative conditions which are necessary and sufficient for one arbitrary state of a system A to be accessible from another state of A in a natural process which involves not only A but also certain reservoirs. It is verified that, when the theory is specialised to equilibrium, the result is the well known partial Legendre transform 'picture' of the Helmholtz and Massieu function representations.

Abstract: A Hamiltonian-based model of many harmonically interacting massive particles that are subject to linear friction and coupled to heat baths at different temperatures is used to study the dynamic approach to equilibrium and nonequilibrium stationary states. An equilibrium system is here defined as a system whose stationary distribution equals the Boltzmann distribution, the relation of this definition to the conditions of detailed balance and vanishing probability current is discussed both for underdamped as well as for overdamped systems. Based on the exactly calculated dynamic approach to the stationary distribution, the functional that governs this approach, which is called the free entropy ${\mathcal{S}}_{\mathrm{free}}(t)$, is constructed. For the stationary distribution ${\mathcal{S}}_{\mathrm{free}}(t)$ becomes maximal and its time derivative, the free entropy production ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\mathcal{S}}}_{\mathrm{free}}(t)$, is minimal and vanishes. Thus, ${\mathcal{S}}_{\mathrm{free}}(t)$ characterizes equilibrium as well as nonequilibrium stationary distributions by their extremal and stability properties. For an equilibrium system, i.e., if all heat baths have the same temperature, the free entropy equals the negative free energy divided by temperature and thus corresponds to the Massieu function which was previously introduced in an alternative formulation of statistical mechanics. Using a systematic perturbative scheme for calculating velocity and position correlations in the overdamped massless limit, explicit results for few particles are presented: For two particles localization in position and momentum space is demonstrated in the nonequilibrium stationary state, indicative of a tendency to phase separate. For three elastically interacting particles heat flows from a particle coupled to a cold reservoir to a particle coupled to a warm reservoir if the third reservoir is sufficiently hot. This does not constitute a violation of the second law of thermodynamics, but rather demonstrates that a particle in such a nonequilibrium system is not characterized by an effective temperature which equals the temperature of the heat bath it is coupled to. Active particle models can be described in the same general framework, which thereby allows us to characterize their entropy production not only in the stationary state but also in the approach to the stationary nonequilibrium state. Finally, the connection to nonequilibrium thermodynamics formulations that include the reservoir entropy production is discussed.

Abstract: A general variational principle concerning non-stationary purely dissipative phenomena, obeying ONSAGER's reciprocal relations, is presented. The principle expresses that the variation of the sum of the total entropy production and the time derivative of the MASSIEU function is equal to zero. It is shown that the EULER-LAGRANGE equations corresponding to arbitrary variations of the state variables yield the conservation laws. Other general criteria for non-stationary processes have been given by GLANSDORFF-PRIGOGINE and GYARMATI. They are compared with our principle. Two illustrative examples are considered: coupled heat and electrical conduction in an isotropic medium and chemical reactions coupled with diffusion of matter and heat conduction in a multi-component fluid.