Systems with negative specific heat

Thirring, Walter

doi:10.1007/bf01403177

Cited by 446 publications

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“…We already mentioned the work of Lynden-Bell and Wood [7], who seem to have been the first to observe such dips in the entropy function of certain gravitational many-body systems (see [11] for a historical account). Similar observations have also been reported in the same context by Thirring and Hertel [12,13], and by Gross [14,15] more recently. For a recent survey of the subject, the reader is invited to consult the comprehensive collection of papers edited by Dauxois et al [16]; it covers a wide range of physical models for which nonconcave anomalies of the microcanonical entropy function have been observed, and contains much information about the physics of these models.…”

Section: Discussionsupporting

confidence: 90%

An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles

Touchette

2004

Physica A: Statistical Mechanics and its Applications

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Section: Discussionsupporting

confidence: 90%

An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles

Touchette

2004

Physica A: Statistical Mechanics and its Applications

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“…Self-gravitating or long-range attractive interacting systems exhibit several peculiar features [1][2][3][4], such as gravothermal catastrophe [5,6], negative specific heat [7][8][9], violent relaxation [10], nonequilibrium thermodynamics, and nonextensive statistical mechanics [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. In these systems, the velocity distributions are non-Gaussian [31][32][33], especially in quasiequilibrium states and metastable states.…”

Section: Introductionmentioning

confidence: 99%

Transition of velocity distributions in collapsing self-gravitatingN-body systems

Komatsu

Kiwata

2012

Phys. Rev. E

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By means of N -body simulations, we study the evolution of gravity-dominated systems from an early relaxation to a collapse, focusing on the velocity distributions and thermodynamic properties. To simulate the dynamical evolution, we consider self-gravitating small N -body systems enclosed in a spherical container with adiabatic or semipermeable walls. It is demonstrated that in the early relaxation process, the velocity distribution is non-Gaussian and q-Gaussian, since the system is in quasiequilibrium states (here q is the Tsallis entropic parameter). Thereafter, the velocity distribution undergoes higher non-Gaussian distributions, especially when the core forms rapidly in the collapse process; i.e., q tends to be larger than that for the quasiequilibrium state, since the velocity distribution further deviates from Gaussian. However, after the core forms sufficiently, the velocity distribution gradually relaxes toward a Gaussian-like distribution. Accordingly, the velocity distribution evolves from a non-Gaussian distribution through a higher non-Gaussian distribution to a Gaussian-like distribution; i.e., the velocity distribution does not monotonically relax toward a Gaussian-like distribution in our collapse simulations. We clearly show such a transition of the velocity distribution, based not only on the Tsallis entropic parameter but also on the ratio of velocity moments. We also find that a negative specific heat occurs in a collapse process with mass and energy loss (such as the escape of stars from globular clusters), even if the velocity distribution is Gaussian-like.

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“…In the canonical ensemble the temperatureβ −1 t corresponds to a discontinuity in the state energy irrespectively of the behavior of the entropy betweenē (1) β andē (2) β . The microcanonical caloric curve in the phase transition region may either converge towards the Maxwell construction or keep a backbending behavior, since a negative heat capacity system can be thermodynamically stable even in the thermodynamical limit if it is isolated [10]. This point has been recently made in somewhat different words by Leyvraz and Ruffo [11].…”

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Failure of thermodynamics near a phase transition

Gulminelli

Chomaz

2002

Phys. Rev. E

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We investigate the relation between various statistical ensembles of finite systems. If ensembles differ at the level of fluctuations of the order parameter, we show that the equations of states can present major differences. A sufficient condition for this inequivalence to survive at the thermodynamical limit is worked out. If energy consists in a kinetic and a potential part, the microcanonical ensemble does not converge towards the canonical ensemble when the partial heat capacities per particle fulfill the relation c −1PACS numbers: 05.20. Gg, 64.10.+h, 05.70.Fh In most textbooks the equivalence between the different statistical ensembles is demonstrated at the thermodynamical limit through the Van Hove theorem [1]. Indeed ensembles differ at the level of fluctuations which are generally believed to induce small corrections in finite systems and to become negligeable at the limit of infinite systems.In this paper we will show that this might not be always the case. For finite systems, two ensembles which put different constraints on the fluctuations of the order parameter lead to qualitatively different equations of states close to a first order phase transition. As an example the microcanonical heat capacity may diverge to become negative while the canonical one remains always positive and finite [2,3]. Such inequivalences may survive at the thermodynamical limit for systems involving long range forces [4,5]. Looking at the general properties of the order parameter distribution a sufficient condition for this behavior to show up can be explitely worked out.Let us first concentrate on finite systems. For simplicity we will consider the microcanonical and the canonical ensemble characterized by the energy E and the temperature β −1 respectively, but our discussion is valid for any couple of conjugated extensive and intensive variables.The microcanonical ensemble is characterized by the level density W (E) and the entropy S = log W .The caloric curve is then T −1 = ∂ E S. The canonical partition sum is the Laplace transform of W : Z β = dEW exp(−βE). In this article we will assume that the partition sum converges; this is not always the case [6] and indeed the impossibility to normalize the distribution W exp(−βE) is already a known case of ensemble inequivalence.In finite systems, the canonical ensemble differs from the microcanonical one since it does not correspond to a unique energy but to a distribution P β (E) = exp(S(E) − βE − log Z β ). If P β has a single maximum the average energy E β = −∂ β ln Z β can also be computed using a saddle point approximation around the most probable energy E βwith g β (x) = c 0 +c 3 x 3 +c 4 x 4 +. . . . If P β is symmetric,The definition of saddle impliesmeaning that the microcanonical caloric curve T (E) exactly coincides with the canonical one β −1 ( E ). However in a finite system the distribution may be not symmetric so that the two curves can be shifted :.. withg β the series of the odd terms of g β . However, the shift δ is in most cases small so that when P β h...

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Systems with negative specific heat

Cited by 446 publications

References 0 publications

An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles

An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles

Transition of velocity distributions in collapsing self-gravitatingN-body systems

Failure of thermodynamics near a phase transition

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