Nonextensive thermodynamic relations

Abstract

The generalized zeroth law of thermodynamics indicates that the physical temperature in nonextensive statistical mechanics is different from the inverse of the Lagrange multiplier, β. This fact leads to modifications of some of thermodynamic relations for nonextensive systems. Here, taking the first law of thermodynamics and the Legendre transform structure as the basic premises, it is found that Clausius' definition of the thermodynamic entropy has to be appropriately changed, and accordingly the thermodynamic relations proposed by Tsallis et al. (Physica A 261 (1998) 534) are also to be modified. It is shown that the definition of specific heat and the equation of state remain form invariant. As an application, the classical gas model is reexamined and, in marked contrast with the previous result obtained by Abe (Phys. Lett. A 263 (1999) 424; Erratum: 267 (2000) 456) using the unphysical temperature and the unphysical pressure, the specific heat and the equation of state are found to be similar to those in ordinary extensive thermodynamics.

Introduction

It is generally believed that the framework of thermodynamics will remain unchanged forever even if the microscopic dynamical laws may be modified in the future. This optimism for robustness of thermodynamics might have its origin in the historical fact that statistical mechanics has been formulated in accordance with thermodynamics. However, if some of basic premises are removed from ordinary Boltzmann–Gibbs statistical mechanics, there is actually no a priori reason any more to be able to expect that all thermodynamic relations still remain unchanged. In such a situation, it is essential to specify a set of physical principles that should be preserved through this generalization.

In this Letter, we discuss how removal of the assumption of extensivity of entropy from statistical mechanics leads to modifications of the thermodynamic relations. We develop this discussion by taking Tsallis' nonextensive statistical mechanics as an example [1], [2], [3]. A point of crucial importance is that, if entropy is nonextensive, the physical temperature is not simply the inverse of the Lagrange multiplier associated with the energy constraint but a variable correctly defined through the generalized zeroth law of thermodynamics [4], [5], [6], [7], [8]. The definition of the physical pressure also becomes different from the ordinary one. Taking into account these facts as well as the first law of thermodynamics and the Legendre transform structure [2], [9], [10], we show that Clausius' definition of the thermodynamic entropy has to be appropriately modified. We also show that the specific heat and the equation of state defined in terms of the physical temperature and the physical pressure remain form invariant. As an application, we reexamine the classical gas model. In marked contrast with the previous result using the “unphysical temperature” and the “unphysical pressure” [11], the specific heat and the equation of state are found to have the same forms as the ordinary ones in extensive thermodynamics.