The Nonequilibrium Thermodynamics of Small Systems

The interactions of tiny objects with their environment are dominated by thermal fluctuations. Guided by theory and assisted by micromanipulation tools, scientists have begun to study such interactions in detail.Comment: PDF file, 13 pages. Long version of the paper published in Physics Toda

Fragmentation phase transition in atomic clusters I --- Microcanonical thermodynamics

Here we first develop the thermodynamics of microcanonical phase transitions of first and second order in systems which are thermodynamically stable in the sense of van Hove. We show how both kinds of phase transitions can unambiguously be identified in relatively small isolated systems of $\sim 100$ atoms by the shape of the microcanonical caloric equation of state $$ I.e. within microcanonical thermodynamics one does not need to go to the thermodynamic limit in order to identify phase transitions. In contrast to ordinary (canonical) thermodynamics of the bulk microcanonical thermodynamics (MT) gives an insight into the coexistence region. The essential three parameters which identify the transition to be of first order, the transition temperature $T_{tr}$, the latent heat $q_{lat}$, and the interphase surface entropy $\Delta s_{surf}$ can very well be determined in relatively small systems like clusters by MT. The phase transition towards fragmentation is introduced. The general features of MT as applied to the fragmentation of atomic clusters are discussed. The similarities and differences to the boiling of macrosystems are pointed out.Comment: Same as before, abstract shortened my e-mail address: [email protected]

Thermodynamics of small Fermi systems: quantum statistical fluctuations

We investigate the probability distribution of the quantum fluctuations of thermodynamic functions of finite, ballistic, phase-coherent Fermi gases. Depending on the chaotic or integrable nature of the underlying classical dynamics, on the thermodynamic function considered, and on temperature, we find that the probability distributions are dominated either (i) by the local fluctuations of the single-particle spectrum on the scale of the mean level spacing, or (ii) by the long-range modulations of that spectrum produced by the short periodic orbits. In case (i) the probability distributions are computed using the appropriate local universality class, uncorrelated levels for integrable systems and random matrix theory for chaotic ones. In case (ii) all the moments of the distributions can be explicitly computed in terms of periodic orbit theory, and are system-dependent, non-universal, functions. The dependence on temperature and number of particles of the fluctuations is explicitly computed in all cases, and the different relevant energy scales are displayed.Comment: 24 pages, 7 figures, 5 table

Thermodynamics of spin systems on small-world hypergraphs

We study the thermodynamic properties of spin systems on small-world hypergraphs, obtained by superimposing sparse Poisson random graphs with p-spin interactions onto a one-dimensional Ising chain with nearest-neighbor interactions. We use replica-symmetric transfer-matrix techniques to derive a set of fixed-point equations describing the relevant order parameters and free energy, and solve them employing population dynamics. In the special case where the number of connections per site is of the order of the system size we are able to solve the model analytically. In the more general case where the number of connections is finite we determine the static and dynamic ferromagnetic-paramagnetic transitions using population dynamics. The results are tested against Monte-Carlo simulations.Comment: 14 pages, 7 figures; Added 2 figures. Extended result

Phase Transitions in "Small" Systems - A Challenge for Thermodynamics

Traditionally, phase transitions are defined in the thermodynamic limit only. We propose a new formulation of equilibrium thermo-dynamics that is based entirely on mechanics and reflects just the {\em geometry and topology} of the N-body phase-space as function of the conserved quantities, energy, particle number and others. This allows to define thermo-statistics {\em without the use of the thermodynamic limit}, to apply it to ``Small'' systems as well and to define phase transitions unambiguously also there. ``Small'' systems are systems where the linear dimension is of the characteristic range of the interaction between the particles. Also astrophysical systems are ``Small'' in this sense. Boltzmann defines the entropy as the logarithm of the area $W(E,N)=e^{S(E,N)}$ of the surface in the mechanical N-body phase space at total energy E. The topology of S(E,N) or more precisely, of the curvature determinant $D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial N^2-(\partial^2S/\partial E\partial N)^2$ allows the classification of phase transitions {\em without taking the thermodynamic limit}. The topology gives further a simple and transparent definition of the {\em order parameter.} Attention: Boltzmann's entropy S(E) as defined here is different from the information entropy and can even be non-extensive and convex.Comment: 8 pages, 4 figures, Invited paper for CRIS200

Quantum Fluctuation Theorems

Recent advances in experimental techniques allow one to measure and control systems at the level of single molecules and atoms. Here gaining information about fluctuating thermodynamic quantities is crucial for understanding nonequilibrium thermodynamic behavior of small systems. To achieve this aim, stochastic thermodynamics offers a theoretical framework, and nonequilibrium equalities such as Jarzynski equality and fluctuation theorems provide key information about the fluctuating thermodynamic quantities. We review the recent progress in quantum fluctuation theorems, including the studies of Maxwell's demon which plays a crucial role in connecting thermodynamics with information.Comment: As a chapter of: F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (eds.), "Thermodynamics in the quantum regime - Fundamental Aspects and New Directions", (Springer International Publishing, 2018