Statistical ensemble

Abstract: A sufficiently fine ferromagnetic particle has a uniform vector magnetization whose magnitude is essentially constant, but whose direction fluctuates because of thermal agitation. The fluctuations are important in superparamagnetism and in magnetic aftereffect. The problem is approached here by methods familiar in the theory of stochastic processes. The "Langevin equation" of the problem is assumed to be Gilbert's equation of motion augmented by a "random-field" term. Consideration of a statistical ensemble of such particles leads to a "Fokker-Planck" partial differential equation, which describes the evolution of the probability density of orientations. The random-field concept, though convenient, can be avoided by use of the fluctuation-dissipation theorem. The Fokker-Planck equation, in general, is complicated by the presence of gyroscopic terms. These drop out in the case of axial symmetry: then the problem of finding nonequilibrium solutions can be restated as a minimization problem, susceptible to approximate treatments. The case of energy barriers large in comparison with $\mathrm{kT}$ is treated both by approximate minimization and by an adaptation of Kramers' treatment of the escape of particles over barriers. The limits of validity of the discrete-orientation approximation are discussed.

Abstract: Thermodynamics, fundamentals conditions for equilibrium and stability statistical mechanics non-interacting (ideal) systems statistical mechanical theory of phase transitions Monte Carlo method in statistical mechanics classical fluids statistical mechanics of non-equilibrium systems.

Abstract: New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail, based mathematically upon the orthogonal, unitary, and symplectic groups. The orthogonal ensemble is relevant in most practical circumstances, the unitary ensemble applies only when time‐reversal invariance is violated, and the symplectic ensemble applies only to odd‐spin systems without rotational symmetry. The probability‐distributions for the energy levels are calculated in the three cases. Repulsion between neighboring levels is strongest in the symplectic ensemble and weakest in the orthogonal ensemble. An exact mathematical correspondence is found between these eigenvalue distributions and the statistical mechanics of a one‐dimensional classical Coulomb gas at three different temperatures. An unproved conjecture is put forward, expressing the thermodynamic variables of the Coulomb gas in closed analytic form as functions of temperature. By means of general group‐theoretical arguments, the conjecture is proved for the three temperatures which are directly relevant to the eigenvalue distribution problem. The electrostatic analog is exploited in order to deduce precise statements concerning the entropy, or degree of irregularity, of the eigenvalue distributions. Comparison of the theory with experimental data will be made in a subsequent paper.

Abstract: 1. Introduction 2. Classical Mechanics 3. Theoretical Foundations of Classical Statistical Mechanics 4. The Microcanonical Ensemble and Introduction to Molecular Dynamics 5. The Canonical Ensemble 6. The Isobaric Ensembles 7. The Grand Canonical Ensemble 8. Monte Carlo Methods in Statistical Mechanics 9. Free Energy Calculations 10. Quantum Mechanics 11. Quantum Ensembles and the Density Matrix 12. Quantum Ideal Gases: Fermi-Dirac and Bose-Einstein Statistics 13. The Feynman Path Integral 14. Classical Time-Dependent Statistical Mechanics and Systems Away from Equilibrium 15. Quantum Time-Dependent Statistical Mechanics 16. The Generalized Langevin Equation 17. Advanced Sampling Approaches 18. Critical Phenomena 19. Conclusions and Perspectives