Microcanonical Specific Heat Research Articles

Negative specific heat with trapped ultracold quantum gases

The second law of thermodynamics normally prescribes that heat tends to disperse, but in certain cases it instead implies that heat will spontaneously concentrate. The spontaneous formation of stars out of cold cosmic nebulae, without which the universe would be dark and dead, is an example of this phenomenon. Here we show that the counter-intuitive thermodynamics of spontaneous heat concentration can be studied experimentally with trapped quantum gases, by using optical lattice potentials to realize weakly coupled arrays of simple dynamical subsystems, so that under the standard assumptions of statistical mechanics, the behavior of the whole system can be predicted from ensemble properties of the isolated components. A naive application of the standard statistical mechanical formalism then identifies the subsystem excitations as heat in this case, but predicts them to share the peculiar property of self-gravitating protostars, of having negative micro-canonical specific heat. Numerical solution of real-time evolution equations confirms the spontaneous concentration of heat in such arrays, with initially dispersed energy condensing quickly into dense ‘droplets’. Analysis of the nonlinear dynamics in adiabatic terms allows it to be related to familiar modulational instabilities. The model thus provides an example of a dictionary mesoscopic system, in which the same non-trivial phenomenon can be understood in both thermodynamical and mechanical terms.

Ensemble Inequivalence in the Spherical Spin Glass Model with Nonlinear Interactions

We investigate the ensemble inequivalence of the spherical spin glass model with nonlinear interactions of polynomial order $p$. This model is solved exactly for arbitrary $p$ and is shown to have first-order phase transitions between the paramagnetic and spin glass or ferromagnetic phases for $p \geq 5$. In the parameter region around the first-order transitions, the solutions give different results depending on the ensemble used for the analysis. In particular, we observe that the microcanonical specific heat can be negative and the phase may not be uniquely determined by the temperature.

Microcanonical Phase Diagram of the BEG and Ising Models

The density of states of long-range Blume—Emery—Griffiths (BEG) and short-range Ising models are obtained by using Wang—Landau sampling with adaptive windows in energy and magnetization space. With accurate density of states, we are able to calculate the microcanonical specific heat of fixed magnetization introduced by Kastner et al. in the regions of positive and negative temperature. The microcanonical phase diagram of the Ising model shows a continuous phase transition at a negative temperature in energy and magnetization plane. However the phase diagram of the long-range model constructed by peaks of the microcanonical specific heat looks obviously different from the Ising chart.

Microcanonical analyses of homopolymer aggregation processes

Using replica-exchange multicanonical Monte Carlo simulation, the aggregates of two homopolymers were numerically investigated through the microcanonical analysis method. The microcanonical entropy showed one convex function in the transition region, leading to a negative microcanonical specific heat. The origin of temperature backbending was the rearrangement of the segments during the process of aggregation; this aggregation process proceeded via a nucleation and growth mechanism. It was observed that the segments with a sequence number from 10 to 13 in the polymer chain have leading effects on the aggregation.

Microcanonical Analyses of Peptide Aggregation Processes

We propose the use of microcanonical analyses for numerical studies of peptide aggregation transitions. Performing multicanonical Monte Carlo simulations of a simple hydrophobic-polar continuum model for interacting heteropolymers of finite length, we find that the microcanonical entropy behaves convex in the transition region, leading to a negative microcanonical specific heat. As this effect is also seen in first-order-like transitions of other finite systems, our results provide clear evidence for recent hints that the characterization of phase separation in first-order-like transitions of finite systems profits from this microcanonical view.

Finite-size behaviour of the microcanonical specific heat

For models which exhibit a continuous phase transition in the thermodynamic limit a numerical study of small systems reveals a non-monotonic behaviour of the microcanonical specific heat as a function of the system size. This is in contrast to a treatment in the canonical ensemble where the maximum of the specific heat increases monotonically with the size of the system. A phenomenological theory is developed which permits to describe this peculiar behaviour of the microcanonical specific heat and allows in principle the determination of microcanonical critical exponents.

Quantumlike short-time behavior of a classical crystal.

We have performed a molecular-dynamics simulation of a face-centered-cubic Lennard-Jones crystal, and studied its relaxation toward equilibrium and its microcanonical equilibrium dynamics through the computation of the normal modes. At low temperature, the weak interaction among normal modes yields a very slow relaxation of the fluctuation of the kinetic energy; this requires a new formulation of the measure of the microcanonical specific heat at constant volume. This specific heat turns out to depend on the time of observation; for times of the order of 20 ps, its values are much nearer to the quantum ones than to the value 3R predicted by the classical Dulong and Petit law. For longer observation times, the classical specific heat progressively approaches 3R over most of the temperature range of the solid crystal, with the exception of the lowest temperature range, where it still drops to values close to zero. The time dependence of the specific heat of the crystal is similar to the behavior found in a supercooled liquid near the glass transition.

Negative specific heat, the thermodynamic limit, and ergodicity.

Thermodynamic stability, in particular, the positivity of the specific heat in the microcanonical ensemble, is not an automatic consequence of the thermodynamic limit. But it holds under special circumstances such as for the most important case of quantum-mechanical Coulomb systems. Therefore, it is surprising that there are experimental indications to the contrary. In this Letter we study a simple model for which the microcanonical specific heat is positive, if the system is ergodic. However, if the system is not ergodic, the energy shell in phase space has some ergodic components with a negative specific heat. This provides another possible general pathway for a negative specific heat in addition to the commonly accepted, the small number of particles.

Gravitational thermodynamics and the generalized second law

We survey the statistical thermodynamics of systems, occurring in astrophysics, whose properties are dominated by gravitational forces. In the non-relativistic case, the lack of extensivity of these systems leads to a non-standard thermodynamics, with a two-phase region in which the microcanonical specific heat is negative. In the relativistic case, there arises the phenomenon of gravitational collapse, in which stars become black holes. We argue that, since these systems have no microstructure that can be observed from outside, the statistical mechanical concepts of microstates and entropy are inapplicable to them. Accordingly, we formulate the thermodynamics of processes, in which a black hole participates, in terms of observable quantities that govern changes in the Gibbs potential of the open system constituting the exterior region. By a simple argument, based only on general demands of quantum theory, relativity and statistical thermodynamics, we infer that the sum of the entropy of the exterior region and a universal constant times the area of the black hole does not decrease in any process. This is precisely the generalized second law, first proposed by Bekenstein on the basis of the hypothesis that a black hole has an entropy proportional to its surface area. By contrast, our derivation of this law does not involve any concept of black hole entropy, since it is based on the statistical thermodynamics of observable open systems.