VSOP19, Quy Nhon 3-18/08/2013
Ngo Van Thanh, Institute of Physics, Hanoi, Vietnam.
I.1. Introduction I.2. Thermodynamics and statistical mechanics I.3. Phase transition I.4. Probability theory
Part I. Basics 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
Monte Carlo Simulation in Statistical Physics: An Introduction
D. Landau and K. Binder, (Cambridge University Press, 2009).
Understanding Molecular Simulation : From Algorithms to Applications
K. Binder and D. W. Heermann (Springer-Verlag Berlin Heidelberg, 2010).
Frustrated Spin Systems
D. Frenkel, (Academic Press, 2002).
Lecture notes PDF files : http://iop.vast.ac.vn/~nvthanh/cours/vsop/ Example code : http://iop.vast.ac.vn/~nvthanh/cours/vsop/code/
H. T. Diep, 2nd Ed. (World Scientific, 2013).
Experiment
I.1. Introduction
Revolution of science :
Nature
The old division of physics into “experimental and “theoretical” branches is not complete.
Computer simulation :
Third branch : COMPUTER SIMULATION
Theory
Simulation
A tool to exploit the electronic
computing machines for the development of nuclear weapons and code breaking
Monte Carlo simulation
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
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
Free energy
(1.2)
Internal Energy
(1.3)
(1.4)
Entropy
Free energy differences
Fluctuations
(1.5)
The probability
The average energy
with
(1.6)
Specific heat
Magnetization
(1.7)
Susceptibility
(1.8)
(1.9)
For a system in a pure phase
(1.11)
Consider the NVT ensemble
(1.10)
(1.12)
Entropy S and the pressure p are the conjugate variables
(1.14)
(1.13)
Fluctuations of extensive variables (like S) scale with the volume Fluctuations of intensive variables (like p) scale with the inverse volume
Gas
I.3. Phase transition
A phase transition is the transformation
of thermodynamic system from one phase (or state of matter) to another
Freezing
Solid Liquid
Order parameter : The order parameter is a quantity which is zero in one phase, and non-zero in the other.
Melting
Ferromagnet : spontaneous magnetization
Liquid–gas : difference in the density
An order parameter may be :
Liquid crystals : degree of orientational order
a scalar quantity
or a multicomponent quantity
Correlation function
Two-point correlation function, space-dependent
(1.15)
: 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
r : spatial distance
(1.18)
define :
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
latent heat
metastable states
TC T TC T
The magnetization is discontinuous at TC
latent heat
TC
Double peak in energy histogram
Ising antiferromagnetic model on the FCC lattice with N = 24
Second order transition
First derivatives are continuous at the critical temperature
F
Internal energy and magnetization are continuous
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
p
Phase diagram is a type of chart used to show conditions at which thermodynamically distinct phases can occur at equilibrium.
critical point
melting curve
Liquid
vaporization curve
Solid
triple point
Vapor
sublimation curve
TC T
Simplified pressure–temperature phase diagram for water
Critical behavior and exponents
The reduced distance from the critical temperature
Critical exponents describe the behaviour of physical quantities near the phase transitions .
(1.19)
(1.20)
For a magnet, the asymptotic expressions are valid only as
(1.21)
(1.22)
: are the “critical exponents”
(1.23)
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
At
(1.25)
(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
(1.28)
At T is near the critical point, the free energy :
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
The free energy of a d-dimensional system near a phase transition
for comparison to the simulations.
: dimensionless coefficients
R : as the interaction range of the model
For the case of a homogeneous system
(1.32)
(1.33)
In equilibrium the free energy must be a minimum
If
Solve the equation
(1.34)
we have
(1.35)
For
Expanding r in the vicinity of TC :
For
, m1 corresponds to the solution above TC
(1.36)
If
Solve the equation
(1.37)
we have
(1.38)
(1.39)
(1.40)
tricritical point appears when
If : the transition is first order
tricritical exponents
I.4. Probability theory
Basic notions: Considering an elementary event with a countable set of random outcomes Suppose this event occurs repeatedly times, we count how often the outcome is observed
Define probabilities for the outcome :
: never occurs;
: it is certain to occur.
(1.41)
From its definition, we conclude that
(1.42)
and are “mutually exclusive” events
the occurrence of implies that does not occur and vice versa.
Consider two events
(1.43)
one with outcomes and probabilities
Defines the outcome and the joint probabilities
the second with outcomes and probabilities
If the events are independent
If they are not independent, one define the conditional probability
(1.44)
(1.45)
Conditional probability that occurs, given that occurs
For the outcome of random events
Defines the expectation value of a random variable
The expectation value of a real function
(1.46)
Consider two functions with the linear combination
Defining the nth moment as
(1.48)
The cumulants
(1.47)
(1.49)
For , the cumulant is called the “variance”
(1.50)
For two random variables
If are independent
(1.51)
Covariance of
(1.53)
(1.52)
As a measure of the degree of independence of the two random variables
Special probability distributions and the central limit theorem Consider a very large number of events
two events are mutually exclusive and exhaustive
Suppose that N independent samples of these events occur
(1.54)
each outcome is either 0 or 1 denote the sum of these outcomes
(1.55)
The binomial distribution
(1.56)
the probability that of the is 1 and is 0
(1.57)
Suppose we have two outcomes (1, 0) of an experiment
if the outcome is 0, the experiment is repeated, otherwise we stop the random variable of interest is the number n of experiments until we get the outcome 1 The geometrical distribution
(1.58)
In the case that the probability of success is very small, we have the Poisson distribution
(1.59)
(1.60)
Gaussian distribution
this is an approximation to the binomial distribution in the case of a very large number of possible outcomes and a very large number of samples.
Statistical errors Suppose the quantity A is distributed according to a Gaussian
mean value and width
consider n statistically independent observations {Ai} of this quantity A unbiased estimator of the mean of this distribution:
(1.61)
the standard error of this estimate:
Consider deviations
the mean square deviation
(1.62)
(1.63)
use the relation
(1.64)
we have
For simple sampling Monte Carlo, the computation of errors of averages
(1.65)
For importance sampling Monte Carlo
(1.66)
: correlation time : the time interval between subsequently generated states
Markov chains and master equations
The concept of Markov chains is so central to Monte Carlo simulations.
Define a stochastic process at discrete times labeled consecutively
Consider a system with a finite set of possible states
Denote by the state the system is in at time ,
consider the conditional probability that
Markov process
if this conditional probability is independent of all states but the immediate predecessor
(1.67)
(1.68)
Markov chain
The corresponding sequence of states the transition probability to move from state i to state j
(1.69)
require that:
The total probability, at time the system is in state
(1.70)
Master equation
(1.71)
Treating time as a continuous variable, we rewrite
(1.72)
This equation describes the balance of gain and loss processes
(1.73)
since the probabilities of the events are “mutually” exclusive so, the total probability for a move away from the state j :
The master equation with the equilibrium probability
(1.74)
the detailed balance with the equilibrium probability
(1.75)
