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We make some simple observations on basic issues pertaining to thermostatistical formalisms.
arXiv (Cornell University), 1999
We show that the latest thermostatistical formalism based on the Havrda & Charvat-Daróczy-Tsallis entropy $S_q [ \rho ] = (q-1)^{-1}(1- tr (\rho^q))$ proposed by Tsallis, Mendes and Plastino is {\em equivalent} to the first one proposed by Tsallis in 1988. Here, equivalent means: {\em the ``equilibrium'' state predicted by either formalism using $q$ leads to the same expectation values for all observables as that predicted by the other formalism using $1/q$''}. We also point out once again that the basic property of {\em transitivity of equilibrium} (e.g., the $0^{th}$ Law of Thermodynamics) fails in these formalisms.
Arxiv preprint cond-mat/0403757, 2004
A realistic and objective axiomatic formulation of Thermostatics for composite systems is presented. The main feature of our axiomatics is that it is free of empirical definitions. In particular, the basic concepts of the theory, such as those of entropy, heat and temperature, are characterized only by the axiomatic basis and the theorems derived from it. We also show that the concept of (quasi)static process does not belong to the body of Thermostatics.
Physica A: Statistical Mechanics and its Applications, 2003
It is shown that a recent proposal to give physically meaningful definitions of temperature and pressure within Tsallis formalism for non-extensive thermostatistics leads to expressions which coincide with those obtained by using the standard Boltzmann formalism of Statistical Mechanics.
Physical Review E, 2019
The class of strongly pseudo-additive entropies, which can be represented as an increasing continuous transformation of Shannon and Rényi entropies, have intensively been studied in previous decades. Although their mathematical structure has thoroughly been explored and established by generalized Shannon-Khinchin axioms, the analysis of their thermostatistical properties have mostly been limited to special cases which belong to two parameter Sharma-Mittal entropy class, such as Tsallis, Renyi and Gaussian entropies. In this paper we present a general analysis of the strongly pseudo-additive entropies thermostatistics by taking into account both linear and escort constraints on internal energy. We develop two types of dualities between the thermostatistics formalisms. By the first one, the formalism of Rényi entropy is transformed in the formalism of SPA entropy under general energy constraint and, by the second one, the generalized thermostatistics which corresponds to the linear constraint is transformed into the one which corresponds to the escort constraint. Thus, we establish the equivalence between four different thermostatistics formalisms based on Rényi and SPA entropies coupled with linear and escort constraints and we provide the transformation formulas. In this way we obtain a general framework which is applicable to the wide class of entropies and constraints previously discussed in the literature. As an example, we rederive maximum entropy distributions for Sharma-Mittal entropy and we establish new relationships between the corresponding thermodynamic potentials. We obtain, as special cases, previously developed expressions for maximum entropy distributions and thermodynamic quantities for Tsallis, Rényi and Gaussian entropies. In addition, the results are applied for derivation of thermostatistical relationships for supra-extensive entropy, which has not previously been considered.
2014
}}), Plastino and Rocca suggest that the divergences inherent to the formulation of nonextensive statistical mechanics can be eliminated {\it {via}} the use of $q$-Laplace transformation which is illustrated for the case of a kinetic Hamiltonian system, the harmonic oscillator. The suggested new formulation raises questions which are discussed in the present comment.
Physics Essays, 2006
Thermodynamic laws are not "equations of motion"; in contrast to the specificity of other fundamental laws of physics, the hallmark of thermodynamic laws is generality. This raises the issue of whether auxiliary assumptions required in applying general thermodynamic laws to particular systems are also of a different nature. Lacking proper understanding of this nature, students of thermodynamics encounter a number of puzzles and misconceptions. A central puzzle is how one makes specific predictions with general laws lacking in specificity. Despite these occasional (but conceptually fundamental) difficulties, it is undeniable that examples of the successful application of thermodynamics abound. This paper argues that all successful applications of thermodynamic laws to particular processes making specific predictions are accompanied by auxiliary components -either explicitly or in tacit form. These are either the usual auxiliary assumptions -or a new kind of auxiliary components, which are idealized (nonempirical) contriving or algorithmic means. This is the radical idea (of the Carnot cycle as an algorithmic process) introduced by Carnot. This central thesis concerning the structure of thermodynamic theory is presented with the support of a good number of evidential examples. In comparison with other physical theories, which are nomothetic sciences, thermodynamic theory reveals its difference in structure and aim as being an atypical nomothetic science with the distinctive hybrid feature of nomothetic sciences and principle-based sciences. Real contriving or algorithmic creations change and evolve with time intrinsically. It is argued that this contriving/historical connection characterizes the nature of engineering knowledge, which is identified as being a historical principle-based science. In comparison to engineering knowledge, biological sciences, it is further suggested, reveal their structural similarity with engineering.
European Journal of Physics, 2013
Several approximations are made to study the microcanonical formalism that are valid in the thermodynamics limit. Usually it is assumed that: 1) Stirling's approximation can be used to evaluate the number of microstates; 2) the surface entropy can be replaced by the volume entropy; 3) derivatives can be used even if the energy is not a continuous variable. And it is also assumed that the results obtained in the microcanonical formalism agree with those from the canonical one. However it is not clear if these assumptions are right for very small systems (10-100 particles). To answer this question, two systems with exact solutions (the Einstein model and the two-level system) have been solved with and without these approximations.
Physics Letters A, 2010
By assuming an appropriate energy composition law between two systems governed by the same non-extensive entropy, we revisit the definitions of temperature and pressure, arising from the zeroth principle of thermodynamics, in a manner consistent with the thermostatistics structure of the theory. We show that the definitions of these quantities are sensitive to the composition law of entropy and internal energy governing the system. In this way, we can clarify some questions raised about the possible introduction of intensive variables in the context of non-extensive statistical mechanics.
The Journal of Chemical Physics, 2007
The laws of thermodynamics provide a clear concept of the temperature for an equilibrium system in the continuum limit. Meanwhile, the equipartition theorem allows one to make a connection between the ensemble average of the kinetic energy and the uniform temperature. When a system or its environment is far from equilibrium, however, such an association does not necessarily apply. In small systems, the regression hypothesis may not even apply. Herein, we show that in small nonequilibrium systems, the regression hypothesis still holds though with a generalized definition of the temperature. The latter must now be defined for each such manifestation.
Archiv für Mathematische Logik und Grundlagenforschung, 1984
EPL (Europhysics Letters), 2010
Recently Abe (arXiv:cond-mat/1005.5110v1) claimed that the q-entropy of nonextensive statistical mechanics cannot be generalized for the continuous variables and therefore can be used only in the discrete case. In this letter, we show that the discrete q-entropy can be generalized to continuous variables exactly in the same manner as Boltzmann-Gibbs entropy, contrary to the claim by Abe, so that q-entropy can be used with discrete as well as continuous variables.
The Journal of Chemical Physics, 2010
In 2005, Bright et al. gave numerical evidence that among the family of time reversible deterministic thermostats known as-thermostats, the conventional = 1 thermostat proposed by Hoover and Evans is the only thermostat that is capable of generating an equilibrium state. Using the recently discovered relaxation theorem, we give a mathematical proof that this is true.
Europhysics Letters (epl), 2004
The definitions of the temperature in the nonextensive statistical thermodynamics based on Tsallis entropy are analyzed. A definition of pressure is proposed for nonadditive systems by using a nonadditive effective volume. The thermodynamics of nonadditive photon gas is discussed on this basis. We show that the Stefan-Boltzmann law can be preserved within nonextensive thermodynamics.
In this paper, a theory for systems in contact with a thermal reservoir is developed. We call such systems "thermostated systems." Interest in these systems has been vigorous for about the last 20 years and is illustrated by the Crooks theorem and the Jarzinski equality, two results for the description of systems in full phase space where all coordinates and momenta may be followed through time. These fundamental results follow from the behavior of Markovian systems in phase space for which path integral expressions can be derived from the Smoluchowski equation valid for Markovian systems. In the Introduction a brief account of this approach is given in which it is stressed that some of the motivation for the approach comes from computer simulations in which at an instant in time all momenta can be reversed in order to study the reversed trajectories. The impossibility of doing this reversal experimentally is discussed and a brief review of the origin of irreversibility in dynamical systems is given. Ultimately this leads to an alternative approach that involves "contraction of the description" and the derivation of a coordinate only picture that is intrinsically non-Markovian. In section I we explain how and why we begin our analysis with the Liouville-Langevin equation. The significance of the abundance of water in biological cells is presented and explains the appropriateness of the Liouville-Langevin equation. In section II the projection operator approach of Zwanzig and Mori, although in an essentially modified form, is developed. This shows why the results ultimately obtained are non-Markovian. In section III boson operator representations for the projection operator approach are derived. This makes the interpretation of the equations easier to grasp and facilitates the analysis. In section IV the intrinsic noncommutivity of the boson operators leads to time ordered exponentials in the general final result, equation ( ), the central result of the paper. Section V contains an elementary check of equation ( ) for the case of no potential energy term in the Liouville equation. In section VI what it means for the result to be non-Markovian is explained. Especially pertinent is the failure of the Smoluchowski equation in the non-Markovian case. Thus the path integral methods used in full phase space are not applicable in the contracted description case. Section VIIa contains an advanced check of equation ( ) for the damped harmonic oscillator. The time evolution of the averaged coordinate is given. In section VIIb analysis of the memory kernel in equation ( ) for the harmonic oscillator is presented in a quite long sequence of steps. The time evolution of the averaged square of the coordinate is derived. In section VIII the results for the harmonic oscillator are combined to form the conditional probability distribution for this non-Markovian, Gaussian example. In section IX non-equilibrium thermodynamics results are discussed. The demarcation between generalized non-equilibrium thermodynamic results governed by the Helmholtz free energy and non-thermodynamic results that feature the non-Markovian character of our general framework is elucidated. Underdamping is the key. Finally, in section X we address the incidence of underdamping in sub-cellular biology.
Physica A: Statistical Mechanics and its Applications, 2003
We calculate the fluctuation of the energy of a system in Tsallis statistics following the finite heat bath canonical ensemble approach. We obtain this fluctuation as the second derivative of the logarithm of the partition function plus an additional term. We also find an explicit expression for the relative fluctuation as related to the number of degrees of freedom of the bath and the composite system.
Universidad Simón Bolívar, 2024
The main objective of this article is to present the laws of thermodynamics with a new perspective that involves a paradigm shift in the form of reasoning that has traditionally been inductive in nature, now presented with deductive characteristics. The laws are presented at the beginning in the way they are believed to be and then the different components that make them up are described.
arXiv: Statistical Mechanics, 2003
The nonextensive statistics based on Tsallis entropy have been so far used for the systems composed of subsystems having same $q$. The applicability of this statistics to the systems with different $q$'s is still a matter of investigation. The actual difficulty is that the class of systems to which the theory has been applied is limited by the usual nonadditivity rule of Tsallis entropy which, in reality, has been established for the systems having same $q$ value. In this paper, we propose a more general nonadditivity rule for Tsallis entropy. This rule, as the usual one for same $q$-systems, can be proved to lead uniquely to Tsallis entropy in the context of systems containing different $q$-subsystems. A zeroth law of thermodynamics is established between different $q$-systems on the basis of this new nonadditivity.
Journal of Mathematical Chemistry, 2000
The birth, raise, development and fortunes of a fundamental theory in thermodynamics, the axiomatic thermodynamics, a creation of Constantin Carathéodory, is thoroughly presented together with a summary of Carathéodory's biography. Axiomatic thermodynamics is centered around some interesting properties of Pfaffian differential equations, which are here introduced and used for some well-known cases in thermodynamics.
Progress of Theoretical Physics Supplement, 2006
We consider the relation between the Boltzmann temperature and the Lagrange multipliers associated with energy average in the nonextensive thermostatistics. In Tsallis' canonical ensemble, the Boltzmann temperature depends on energy through the probability distribution unless q = 1. It is shown that the so-called 'physical temperature' introduced in [Phys. Lett. A 281 (2001), 126] is nothing but the ensemble average of the Boltzmann temperature.
Physica A: Statistical Mechanics and its Applications, 2012
We study a system of interacting particles in the framework of the two-parameter Sharma-Mittal entropy S q,r . The two-body Hamiltonian describing the interaction between the particles of the system is obtained requiring that the equilibrium distribution can be factorized in the product of the distributions of the single particles. In the present picture, we derive, according to the zeroth principle of thermodynamics, a possible definition of temperature and pressure and investigate their scaling properties. We show that, if the parameter q scales according to the law q ′ -1 = λ (q -1), temperature and pressure are intensive-like quantities, while entropy has an extensive-like behavior.
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