On statistics and thermodynamics V

The paper deals with the statistical characterization of thermal equilibrium of a system on the basis of the statistical theory of information, estimation and the phenomenological principles of classical thermodynamics.     

On statistics and thermodynamics II

The paper is an attempt on the foundation of statistical thermodynamics on the basis of the thermodynamic theory of measurement.The author wishes to thank U.G.C. for financial assistance.     

On statistics and thermodynamics IV

The paper deals with a statistical analysis of thermal equilibrium in the light of the statistical theory of information and estimation.     

On statistics and thermodynamics I

The paper is an attempt to develop the statistics of thermal equilibrium by incorporating fluctuation in the thermodynamic theory of measurement.The author wishes to thank U.G.C. for financial assistance. He also thanks the referee for his valuable suggestions in revising the paper.     

A generalized thermodynamics for power-law statistics

We show that there exists a natural way to define a condition of generalized thermal equilibrium between systems governed by Tsallis thermostatistics, under the hypotheses that (i) the coupling between the systems is weak, (ii) the structure functions of the systems have a power-law dependence on the energy. It is found that the q values of two such systems at equilibrium must satisfy a relationship involving the respective numbers of degrees of freedom. The physical properties of a Tsallis distribution can be conveniently characterized by a new parameter η which can vary between 0 and +∞, these limits corresponding, respectively, to the two opposite situations of a microcanonical distribution and of a distribution with a predominant power-tail at high energies. We prove that the statistical expression of the thermodynamic functions is univocally determined by the requirements that (a) systems at thermal equilibrium have the same temperature, (b) the definitions of temperature and entropy are consistent with the second law of thermodynamics. We find that, for systems satisfying the hypotheses (i) and (ii) specified above, the thermodynamic entropy is given by Rényi entropy.     

Microcanonical ensemble extensive thermodynamics of Tsallis statistics

The microscopic foundation of the generalized equilibrium statistical mechanics based on the Tsallis entropy is given by using the Gibbs idea of statistical ensembles of the classical and quantum mechanics. The equilibrium distribution functions are derived by the thermodynamic method based upon the use of the fundamental equation of thermodynamics and the statistical definition of the functions of the state of the system. It is shown that if the entropic index ξ=1/(q−1) in the microcanonical ensemble is an extensive variable of the state of the system, then in the thermodynamic limit the principle of additivity and the zero law of thermodynamics are satisfied. In particular, the Tsallis entropy of the system is extensive and the temperature is intensive. Thus, the Tsallis statistics completely satisfies all the postulates of the equilibrium thermodynamics. Moreover, evaluation of the thermodynamic identities in the microcanonical ensemble is provided by the Euler theorem. The principle of additivity and the Euler theorem are explicitly proved by using the illustration of the classical microcanonical ideal gas in the thermodynamic limit.     

Incomplete nonextensive statistics and the zeroth law of thermodynamics

On the basis of the entropy of incomplete statistics (IS) and the joint probability factorization condition, two controversial problems existing in IS are investigated: one is what expression of the internal energy is reasonable for a composite system and the other is whether the traditional zeroth law of thermodynamics is suitable for IS. Some new equivalent expressions of the internal energy of a composite system are derived through accurate mathematical calculation. Moreover, a self-consistent calculation is used to expound that the zeroth law of thermodynamics is also suitable for IS, but it cannot be proven theoretically. Finally, it is pointed out that the generalized zeroth law of thermodynamics for incomplete nonextensive statistics is unnecessary and the nonextensive assumptions for the composite internal energy will lead to mathematical contradiction.     

Statistical thermodynamics of block copolymers: Network statistics

The swelling behavior of a lamellar domain system of an ABA block copolymer is treated, with account taken of the relative population of the B chains as “loops” and “bridges.” The swelling behavior is not a sensitive function of these configurations, since both types in condensed systems are elastically effective, in contrast to the behavior of isolated chains.     

On thermodynamics and gravity

Physics of Atomic Nuclei - A new entropic gravity inspired derivation of general relativity from thermodynamics is presented. This generalizes the “Thermodynamics of Spacetime” approach...     

On relativistic thermodynamics

The thermodynamics of moving bodies is developed from first principles. To do this, it is necessary to augment the laws of thermodynamics with a new principle, which asserts the impossibility of thermal equilibrium between bodies in relative motion. Clausius' theorem is generalized to heat flow between moving systems, and leads naturally to the identification of heat and temperature as Lorentz scalars. The formulation of relativistic statistical mechanics is carried out and the correspondence with classical quantities is made. The quantum distribution laws are generalized to the relativistic case, and are found to differ from their accepted relativistic form.     

On phantom thermodynamics

The thermodynamic properties of dark energy fluids described by an equation of state parameter ω=p/ρ$\omega =p/\rho$ are rediscussed in the context of FRW type geometries. Contrarily to previous claims, it is argued here that the phantom regime ω<−1$\omega <-1$ is not physically possible since that both the temperature and the entropy of every physical fluids must be always positive definite. This means that one cannot appeal to negative temperature in order to save the phantom dark energy hypothesis as has been recently done in the literature. Such a result remains true as long as the chemical potential is zero. However, if the phantom fluid is endowed with a non-null chemical potential, the phantom field hypothesis becomes thermodynamically consistent, that is, there are macroscopic equilibrium states with T>0$T>0$ and S>0$S>0$ in the course of the Universe expansion.     

Impact of loop statistics on the thermodynamics of RNA folding

Loops are abundant in native RNA structures and proliferate close to the unfolding transition. By including a statistical weight approximately l(-c) for loops of length l in the recursion relation for the partition function, we show that the heat capacity depends sensitively on the presence and value of the exponent c, even for a short explicit tRNA sequence. For long homo-RNA, we analytically calculate the critical temperature and critical exponents which exhibit a nonuniversal dependence on c.     

On the thermodynamics of surfaces

Isotropic macroscopic thermodynamics is used to treat a classical model of surfaces. An equation of the Gibbs-Duhem type is derived from which the relation between surface stress and specific surface free energy follows. This derivation specifically considers the pressure dependence on size of a spherical particle bounded by an isotropic surface and hence is not restricted to the usual constant pressure assumptions. It follows from the conditions of thermodynamic equilibrium and the above Gibbs-Duhem relation that surface stress is the fundamental parameter controlling the internal hydrostatic pressure of isotropic particles as well as equilibrium vapor pressure and other phenomena dependent on internal pressure.     

On the relationship between fluctuating irreversible thermodynamics and “extended” irreversible thermodynamics

The relationship between fluctuating irreversible thermodynamics and theories of irreversible processes which include the thermodynamic fluxes as independent variables is explored. It is shown that the usual fluctuating linear theory of irreversible thermodynamics is a contraction of the extended theory. This contraction contains non-Markovian effects dependent upon the relaxation times associated with the thermodynamic fluxes. In the limit that these relaxation times are small, the extended theory is shown to be equivalent to the usual fluctuating thermodynamic theory. A critique of the extended theories is given from the point of view of the mechanistic statistical theory of irreversible processes.     

On thermodynamics near a steady state

The statistical deduction of a stability criterion for steady states (a generalization of the Glansdorff-Prigogine criterion) given in a previous paper is extended in some respects. Dynamical coefficients for the motion of small deviations from a steady state are expressed by correlation functions. The stability criterion is applied to some models of chemical reactions and to a model for second break-down in semiconductors.     

On phantom thermodynamics with negative temperature

We discuss the thermodynamic properties of the Friedmann–Robertson–Walker universe with dark energy fluids labelled by ω=p/ρ<−1/3$\omega =p/\rho <-1/3$. Using the integrability condition, we show that the phantom phase of ω<−1$\omega <-1$ can still be thermodynamically allowed even when the temperature takes on negative values because in that case, there exists at least a condition of keeping physical values for p and ρ.     

On the second law of thermodynamics

The second law of thermodynamics has two distinct aspects to its foundations. The first concerns the question of why entropy goes up in the future, and the second, of why it goes down in the past. Statistical physicists tend to be more concerned with the first question and with careful considerations of definition and mathematical detail. The second question is of quite a different nature; it leads into areas of cosmology and quantum gravity, where the mathematical and physical issues are ill understood.