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

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

VSOP19, 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 theory Part 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 methods Part 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 techniques
I.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 volume
I.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 discontinuous F U metastable states metastable latent states heat TC T TC T
 The magnetization is discontinuous at TC  Double peak in energy histogram latent heat TC Ising antiferromagnetic model on the FCC lattice with N = 24
 Second order transition  First derivatives are continuous at the critical temperature  Internal energy and magnetization are continuous F U critical point critical point TC T TC T
 Internal energy and magnetization are continuous  Changes the curvature at the critical temperature TC Ising ferromagnetic model on the FCC lattice with N = 24
Phase diagrams  Phase diagram is a type of chart used to show conditions at which thermodynamically distinct phases can occur at equilibrium. p melting critical point curve Liquid triple vaporization Solid point curve Vapor sublimation curve TC T Simplified pressure–temperature phase diagram for water
Critical behavior and exponents  Critical exponents describe the behaviour of physical quantities near the phase transitions .  The reduced distance from the critical temperature (1.19)  For a magnet, the asymptotic expressions are valid only as (1.20) (1.21) (1.22) (1.23) : are the “critical exponents”  At , for a ferromagnet (1.24)
 For a system in d-dimensions, above the critical temperature the two-body correlation function has the Ornstein–Zernike form (1.25)  At (1.26)  Two-dimensional Ising square lattice  logarithmic divergence of the specific heat  Two-dimensional XY-models The correlation length (1.27) Kosterlitz and Thouless phase transition
Universality and scaling  Consider a simple Ising ferromagnet in a small magnetic field H  At T is near the critical point, the free energy : (1.28) the gap exponent is “scaled” variable  The correlation function is written in scaling form (1.29)  The Rushbrooke equality (1.30)  The “hyperscaling” expression for the lattice dimensionality (1.31)
Landau theory  for comparison to the simulations.  The free energy of a d-dimensional system near a phase transition (1.32) : dimensionless coefficients R : as the interaction range of the model  For the case of a homogeneous system (1.33)
In equilibrium the free energy must be a minimum  If  Solve the equation (1.34) we have (1.35)  Expanding r in the vicinity of TC : For (1.36) For , m1 corresponds to the solution above TC