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Unformatted text preview: Physics 510: Statistical Physics (Winter 2004) Contents I. Fundamentals of Statistical Mechanics 2 A. Review of Hamiltonian Mechanics 2 B. Ensembles and Averages 3 C. Microcanonical ensemble 3 D. Statistical Weight 4 E. Entropy and Temperature 5 F. Canonical Ensemble 7 1. Gibbs distribution 7 2. Free Energy and its Derivatives 8 3. Equipartition theorem 9 4. Gibbs paradox 10 G. Other Ensembles 10 H. Summary 11 I. Review of Thermodynamics 12 J. Fluctuations of basic thermodynamic variables 13 II. Selected Applications 15 A. Van der Waals gas 15 1. VaporLiquid transition 16 2. Fluctuations and stability 16 B. Lattice Gas 17 C. Quantum Statistics 18 1. Bose Condensation 18 2. Ideal Fermi Gas 18 3. Photon and Phonon Gases 19 III. Phase Transitions 21 A. Ising Model 21 B. Landau Theory 22 C. Fluctuations of order parameter 22 D. Zero modes and BerezinskyKosterlitzThouless transition 24 IV. Non-equilibrium statistical physics 25 A. Brownian motion 25 1. Lagevin equation 25 2. Fluctuation-Dissipation Theorem 25 3. Motion in a non-uniform potential 26 B. Linear Response 26 C. Di f usion and Fokker-Planck Equation 26 2 I. FUNDAMENTALS OF STATISTICAL MECHANICS Equilibrium Statistical Mechanics allows us to construct the thermodynamic description of a system starting with classical or quantum mechanics. The fundamental (mechanical) level is de f ned by the Hamiltonian of the system. On the macroscopic level, we are interested in the equations of state , i.e. relationships between thermodynamic observables, such as temperature, density, magnetization, etc. Strictly speaking, the relationship is statistical, not deterministic: the equation of state determines the most likely values of the observable. Statistical mechanics also allows one to describe the F uctuations around this most probable state. Non-equilibrium Statistical Mechanics. The fundamental level is the same as in the previous case (classi- cal/quantum mechanics). On the macroscopic level, one is interested in time evolution of the thermodynamic parameters, for a given initial conditions and/or time-dependent perturbation. "Postmodern" Statistical Mechanics (aka Complex Systems) uses the ideas of statistical physics in the context of non-Hamiltonian systems. Examples: modelling of granular materials, epidemics, economics, social phenomena. A. Review of Hamiltonian Mechanics Microscopic variables : n generalized coordinates q ( q 1 , q 2 , ..., q n ) and momenta p ( p 1 , ...., p n ) . Their time evolution is deterministic and given by Hamiltonian of the system H ( q , p ) through Hamiltons equations of motion: q i = H ( q , p ) p i ; (1) p i = H ( q , p ) q i . (2) Phase Space is the 2 n dimensional set of all possible coordinates and momenta ( q , p ) . In this space, the current position ( q , p ) completely determines the future (and past) evolution of the system. Therefore, phase space trajectories never cross....
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