Lecture Monte carlo simulations: Application to lattice models: Part I - TS. Ngô Văn Thanh

Lecture Monte carlo simulations: Application to lattice models, part I - Basics. The main contents of this chapter include all of the following: Introduction, thermodynamics and statistical mechanics, phase transition, probability theory.
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Lecture Monte carlo simulations: Application to lattice models: Part I - TS. Ngô Văn ThanhVSOP19, Quy Nhon 3-18/08/2013 Ngo Van Thanh, Institute of Physics, Hanoi, Vietnam.Part I. Basics I.1. Introduction I.2. Thermodynamics and statistical mechanics I.3. Phase transition I.4. Probability theoryPart II. Monte Carlo Simulation Methods II.1. The spin models II.2. Boundary conditions II.3. Simple sampling Monte Carlo methods II.4. Importance sampling Monte Carlo methodsPart III. Finite size effects and Reweighting methods III.1. Finite size effects III.2. Single histogram method III.3. Multiple histogram method III.4. Wang-Landau method III.5. The applications A Guide to Monte Carlo Simulations in Statistical Physics D. Landau and K. Binder, (Cambridge University Press, 2009). Monte Carlo Simulation in Statistical Physics: An Introduction K. Binder and D. W. Heermann (Springer-Verlag Berlin Heidelberg, 2010). Understanding Molecular Simulation : From Algorithms to Applications D. Frenkel, (Academic Press, 2002). Frustrated Spin Systems H. T. Diep, 2nd Ed. (World Scientific, 2013). Lecture notes PDF files : http://iop.vast.ac.vn/~nvthanh/cours/vsop/ Example code : http://iop.vast.ac.vn/~nvthanh/cours/vsop/code/I.1. Introduction Experiment Revolution of science :  The old division of physics into “experimental and “theoretical” branches is not complete. Nature  Third branch : COMPUTER SIMULATION Computer simulation :  A tool to exploit the electronic Theory Simulation computing machines for the development of nuclear weapons and code breaking  Developed during and after Second World War  Molecular dynamic simulation and Monte Carlo simulation (50th decade)  Simulation : Perform an experiment on the computer or test a theory Monte Carlo simulation  First review of the use of Monte Carlo simulations (Metropolis and Ulam, 1949)  Use a complete Hamiltonian without any approximative techniquesI.2. Thermodynamics and statistical mechanics Basic notations : Partition function (1.1) the summation is taken over all the possible states of the system : Hamiltonian : Boltzmann constant : Temperature  Probability (1.2) Free energy (1.3) Internal Energy (1.4)  Free energy differences Entropy (1.5) Fluctuations  The probability  The average energy (1.6) with Specific heat (1.7) Magnetization (1.8) Susceptibility (1.9)  For a system in a pure phase (1.10) (1.11) Consider the NVT ensemble  Entropy S and the pressure p are the conjugate variables (1.12) (1.13) (1.14) Fluctuations of extensive variables (like S) scale with the volume Fluctuations of intensive variables (like p) scale with the inverse volumeI.3. Phase transition Gas  A phase transition is the transformation of thermodynamic system from one phase (or state of matter) to another Order parameter : The order parameter is a quantity which is zero in one phase,  Freezing Solid Liquid and non-zero in the other. Melting   Ferromagnet : spontaneous magnetization  Liquid–gas : difference in the density  Liquid crystals : degree of orientational order An order parameter may be :  a scalar quantity  or a multicomponent quantity Correlation function Two-point correlation function, space-dependent (1.15)  r : spatial distance   : the quantity whose correlation is being measured. Time correlation function between two quantities (1.16) If A = B, autocorrelation function of A (1.17) Fluctuations in the quantity A  define : (1.18) First order & second order transition Consider a system which is in thermal equilibrium and undergo a phase transition between a disordered state and one First order transition:  The first derivatives of the free energy are discontinuous at TC  The internal energy is discontinuousF ...