Category: Equilibrium

Foundations of statistical mechanics for unstable interactions

Post author By R Hilfer
Post date May 27, 2021

R. Hilfer

Physical Review E 105, 024142 (2022)
https://doi.org/10.1103/PhysRevE.105.024142

submitted on
Thursday, May 27, 2021

Traditional Boltzmann-Gibbs statistical mechanics does not apply to systems with unstable interactions, because for such systems the conventional thermodynamic limit does not exist. In unstable systems the ground state energy does not have an additive lower bound, i.e., no lower bound linearly proportional to the number N of particles or degrees of freedom. In this article unstable systems are studied whose ground state energy is bounded below by a regularly varying function of N with index \sigma\geq 1. The index \sigma\geq 1 of regular variation introduces a classification with respect to stability. Stable interactions correspond to σ = 1. A simple example for an unstable system with σ =2 is an ideal gas with a nonvanishing constant two-body potential. The foundations of statistical physics are revisited, and generalized ensembles are introduced for unstable interactions in such a way that the thermodynamic limit exists. The extended ensembles are derived by identifying and postulating three basic properties as extended foundations for statistical mechanics: first, extensivity of thermodynamic systems, second, divisibility of equilibrium states, and third, statistical independence of isolated systems. The traditional Boltzmann-Gibbs postulate, resp. the hypothesis of equal a priori probabilities, is identified as a special case of the extended ensembles. Systems with unstable interactions are found to be thermodynamically normal and extensive. The formalism is applied to ideal gases with constant many-body potentials. The results show that, contrary to claims in the literature, stability of the interaction is not a necessary condition for the existence of a thermodynamic limit. As a second example the formalism is applied to the Curie-Weiss-Ising model with strong coupling. This model has index of stability σ = 2. Its thermodynamic potentials [originally obtained in R. Hilfer, Physica A 320, 429 (2003)] are confirmed up to a trivial energy shift. The strong coupling model shows a thermodynamic phase transition of order 1 representing a novel mean-field universality class. The disordered high temperature phase collapses into the ground state of the system. The metastable extension of the high temperature free energy to low temperatures ends at absolute zero in a phase transition of order 1/2. Between absolute zero and the critical temperature of the first order transition all fluctuations are absent.

For more information see

url

pdf

Equilibrium Statistical Physics

Ground state collapse at strong coupling

Post author By R Hilfer
Post date May 26, 2017

R. Hilfer

Journal MESA 8, 307-310 (2017)

submitted on
Friday, May 26, 2017

The infinite range Ising model is usually investigated in the weak coupling limit. Here the model is solved with ferromagnetic coupling at fixed and finite strength. Exact analytical expressions are found for the thermodynamic potentials as functions of enthalpy and external field. These results differ from the potentials for the weak coupling limit. The model shows a temperature driven first order phase transition from a paramagnetic phase at high temperatures into a low temperature phase from which thermal fluctuations are absent.

For more information see

url

Tags 2017, Ising model

Equilibrium Fractional Calculus Statistical Physics

Fractional Calculus and Regular Variation in Thermodynamics

Post author By R Hilfer
Post date May 5, 1999

R. Hilfer

in: Applications of Fractional Calculus in Physics
edited by: R. Hilfer
World Scientific, Singapore, 429-463 (2000)
https://doi.org/10.1142/3779
ISBN: 978-981-02-3457-7

submitted on
Wednesday, May 5, 1999

For more information see

url

Tags regular variation, scaling

Critical phenomena Equilibrium Simulations

Phase Transitions in Dense Lipid Monolayers Grafted to a Surface: Monte Carlo Investigation of a Coarse-Grained Off-Lattice Model

Post author By R Hilfer
Post date April 12, 1996

F. M. Haas, R. Hilfer, K. Binder

The Journal of Physical Chemistry 100 (37), 15290-15300 (1996)
DOI: 10.1021/jp9610980

submitted on
Friday, April 12, 1996

Semiflexible amphiphilic molecules end-grafted at a flat surface are modeled by a bead-spring chain with stiff bond angle potentials. Constant density Monte Carlo simulations are performed varying temperature, density, and chain length of the molecules, whose effective monomers interact with Lennard-Jones potentials. For not too large densities and low temperatures the monolayer is in a quasi-two-dimensional crystalline state, characterized by uniform tilt of the (stretched) chains. Raising the temperature causes a second-order transition into a (still solid) phase with no tilt. For the first time, finite size scaling concepts are applied to a model of a surfactant monolayer, and it is found that the technique in this case again is useful to locate the transition more precisely. For comparison, also a one-dimensional version of the model is studied, and directions for future extensions of this modeling are discussed.

For more information see

url

pdf

Tags lipid layer

Critical phenomena Equilibrium Statistical Physics

Are Critical Finite Size Scaling Functions Calculable From Knowledge of an Appropriate Critical Exponent ?

Post author By ICP
Post date December 6, 1994

R. Hilfer, N.B.Wilding

J. Phys. A: Math. Gen. 28, L281 (1995)
10.1088/0305-4470/28/10/001

submitted on
Tuesday, December 6, 1994

Critical finite size scaling functions for the order parameter distribution of the two and three dimensional Ising model are investigated. Within a recently introduced classification theory of phase transitions the universal part of critical finite size scaling functions has been derived by employing a scaling limit which differs from the traditional finite size scaling limit. In this paper the analytical predictions are compared with Monte Carlo simulation results. We find good agreement between the analytical expression and the simulation results. The agreement is consistent with the possibility that the functional form of the critical finite size scaling function for the order parameter distribution is determined uniquely by only a few universal parameters, most notably the equation of state exponent.

For more information see

url

pdf

html

Tags Ising model, order parameter, scaling function

Critical phenomena Equilibrium Simulations Statistical Physics

Continuum Monte-Carlo Simulations of Phase Transitions in Rodlike Molecules at Surfaces

Post author By R Hilfer
Post date October 28, 1994

R. Hilfer, F.M. Haas, K. Binder

Il Nuovo Cimento D 16, 1297-1303 (1994)
https://doi.org/10.1007/BF02458816

submitted on
Friday, October 28, 1994

Stiff rod-like chain molecules with harmonic bond length potentials and trigonometric bond angle potentials are used to model Langmuir monolayers at high densities. One end of the rod-like molecules is strongly bound to a flat two-dimensional substrate which represents the air-water interface. A ground-state analysis is performed which suggests phase transitions between phases with and without collective uniform tilt. Large-scale off-lattice Monte Carlo simulations over a wide temperature range show in addition to the tilting transition the presence of a strongly constrained melting transition at high temperatures. The latter transition appears to be related to two-dimensional melting of the head group lattice. These findings show that the model contains both, two- and three-dimensional ergodicity breaking solidification transitions. We discuss our findings with respect to experiment.

For more information see

url

pdf

Tags lipid layer

Critical phenomena Equilibrium Simulations Statistical Physics

Layers of Semiflexible Chain Molecules Endgrafted at Interfaces: An Off-Lattice Monte Carlo Simulation

Post author By R Hilfer
Post date July 11, 1994

F.M. Haas, R. Hilfer, K. Binder

Journal of Chemical Physics 102, 2960-2969 (1995)
https://doi.org/10.1063/1.468604

submitted on
Monday, July 11, 1994

A coarse‐grained model for surfactant chain molecules at interfaces in the high density regime is studied using an off‐lattice Monte Carlo technique. The surfactant molecules are modeled as chains consisting of a small number (e.g., seven) of effective monomers. For the modeling of lipid monolayers, each effective monomer is thought to represent several CH2 groups of the alkane chain, but applications of the model to other polymers end grafted at solid surfaces also should be possible. The head segments are restricted to move in the adsorption plane, but otherwise do not differ from the effective monomers, which all interact with Lennard‐Jones potentials. Bond angle and bond length potentials take into account chain connectivity and chain stiffness. The advantage of this crude model is that its phase diagram can be studied in detail. Temperature scans show two phase transitions, a tilting transition at low temperatures between a tilted and an untilted phase, and a melting transition at high temperatures where the lattice of head groups loses its crystalline order.

For more information see

url

pdf

Tags lipid layer

Critical phenomena Equilibrium Statistical Physics

Thermodynamic Scaling Derived via Analytic Continuation from the Classification of Ehrenfest

Post author By R Hilfer
Post date April 2, 1991

R. Hilfer

Physica Scripta 44, 321 (1991)
10.1088/0031-8949/44/4/002

submitted on
Tuesday, April 2, 1991

Scaling of the free energy is derived from thermodynamic arguments. The free energy is found to obey a new generalized scaling form which contains standard scaling as a special case. Contrary to standard scaling the new scaling form permits also nonuniversal exponents. The results are obtained by analytic continuation from the classification scheme of Ehrenfest.

For more information see

url