ACADEMIC RESOURCE
Introduction to Statistical Physics
Statistical Physics
Physics
67 pages · professionally formatted edition
STUDY · REVIEW · REFERENCE
, Introduction to Statistical Physics
Johar M. Ashfaque
1
STATISTICAL PHYSICS 1
, 1. Introduction scope of the theory and leaves many interesting properties
of nature unexplained.
All matter we see around us consists of vast number of
microscopic particles that are in constant motion, and in The idea that the macroscopic properties of matter are
interaction with one another. At any given instant of time, caused by the microscopic constituents was first introduced
it is possible in principle to find the state of this kind of in the second half of the 19th century by James Clerk
many-body system. The laws of physics are such that the Maxwell, Ludwig Boltzmann, Max Planck, Rudolf Clau-
future evolution of the state can then be solved from the sius, and J. Willard Gibbs. The theory was completed in
governing equations of motion. However in practise, the the first half of 20th century by the advent of quantum
number of the constituent particles is typically of the order mechanics that gives the correct description of atomic size
of the Avogadro’s number particles. By applying the probabilistic ideas to systems
in equilibrium, one can obtain all results of the thermody-
NA = 6 · 1023 , namics together with the ability to relate the macroscopic
parameters of the system to its microscopic constituents.
leading to stupendously large set of coupled equations.
This powerful method will be subject in the first half of
Even the storing of the initial state of these equations
this course. The description of systems not in equilibrium
would exceed in multitude the combined memory of all
is much more difficult task. One can still make some gen-
of the computers in the world. This is the so-called many-
eral statements of such systems which leads to statistical
body problem of physics.
physics of irreversible processes. This will be discussed
When the number of the constituent particles is im- briefly in the latter part of the course.
mense, it has turned out useful to use the tools of probabil-
Statistical physics can, in principle, be applied to
ity calculus to describe their emergent behaviour. In this
any state of matter: solids, liquids, gases, matter com-
course, we will not be interested in the details of individ-
posed of several phases/or several components, matter
ual particles, but in the relations between the macroscopic
under extreme conditions of temperature and pressure,
properties that result. In fact, we lack most of the informa-
matter in equilibrium with radiation, and so on. In
tion required to describe the internal state of the system,
addition, the concepts and methods of statistical physics
and therefore our approach has to be probabilistic in na-
have turned out useful in many other fields of science:
ture. This is the starting point of statistical physics.
chemistry, study of dynamical systems, communications,
The equations governing the macroscopic properties (e.g. bioinformatics, complexity, and even stock markets.
energy, pressure, temperature) of a many-body system are
statistical averages of those of the constituent particles.
Therefore, it is implicitly assumed that the averaged quan-
tities experience statistical deviations.
For example, the familiar equation of state of the ideal “If a million monkeys typed ten hours a day, it is
gas extremely unlikely that their output would exactly equal
P V = nRT all the books of the richest libraries of the world; and yet,
relates the average pressure P and average volume V to in comparison, it is even more unlikely that the laws of
the average temperature T . Quite generally, the deviations statistical mechanics would ever be violated, even briefly.”
from the√average in a many-body system with N particles
go as 1/ N . The astounding number of particles, which - Emile Borel, 1913
makes the calculations of the macroscopic properties from
first principles practically impossible, leads to such an ac-
curacy of the statistical results that they can be taken as
exact physical laws.
Motivation
The systematic studies of macroscopic systems started
from the phenomenological studies in the 18th and 19th
centuries. These were done by the likes of James Watt,
William Rankine, Rudolf Clausius, Sadi Carnot, and
William Thomson. The resulted theory of thermodynam-
ics is concerned on heat and its relation to other forms of
energy and work. The strength of thermodynamics is in
its generality, emerging from the fact that the theory does
not depend on any detailed assumptions on the microscopic
properties of the system. This is also the main weakness
of thermodynamics, since only relatively few statements
can be made on such general grounds. This restricts the
2
STATISTICAL PHYSICS 2
, 2. Background performs successive steps, each of which has a length l and
is taken on a random direction. The successive steps are
2.1 Ensemble and Probability statistically independent, so we can denote
The laws of Physics are deterministic in a sense that p = probability of a right step
given the state of a system at one time one can calculate
the state at all later times, by using the time-dependent q = 1 − p = probability of a left step
Schrödinger equation or classical mechanics. Then, the
One obtains a great insight by considering the number p as
outcome of an experiment on the system should be deter-
the fraction of right steps in the ensemble of mental copies
mined completely by this final state. Even though this can
of single steps.
be done in principle, it is practically impossible to deter-
mine an initial state of a system with NA particles, let The random walk of N individual steps is formed by tak-
alone to calculate its time evolution. ing a subset, i.e. a sample, from the ensemble and calcu-
lating the sum of its members. After N steps, the particle
Statistical physics circumvents this problem by consid-
lies at
ering a (infinitely) large set of similar experiments, instead
x = ml,
of a single one. This kind of set of mental copies of similar
systems is called an ensemble 1 . where
Let then S denote a general system and consider an ex- m = nr − nl (2)
periment on the system with a general outcome X. We and nr (nl ) denotes the number of steps to right (left).
denote with Σ the ensemble of mental copies of S. Now, Naturally,
the probability of observing the outcome X is given by −N ≤ m ≤ N
Ω(X) and
P (X) = lim , (1)
Ω(Σ)→∞ Ω(Σ) N = nr + nl . (3)
where Ω(Σ) is the number of systems in the ensemble and
Ω(X) is the number of systems exhibiting the outcome X. By taking into account the number of different ways tak-
This is the scientific definition of probability. ing nr steps out of N , we arrive at the probability
N ! nr n l
Example: Throwing a dice WN (nr ) = p q (4)
nr !nl !
One could in principle determine the initial position and
of taking nr steps to the right. Probability function (4) is
orientation of a dice. Also, one could figure out exactly how
referred to as the binomial distribution since it resembles
the dice is thrown out of hand. By using these as initial
the terms in the binomial expansion.
conditions, one could use the classical equations of motion
to determine exactly which of the six numbers turns up. By using Eqs. (2) and (3), one can show (exercise) that
Practically, this is impossible. the probability of finding a particle at position m after N
steps is
Instead, we usually consider a large group of (nearly)
similarly thrown dices. The result of a single throw is PN (m) = WN (nr )
not anymore deterministic, since we do not have the ex-
N! (5)
act knowledge of the initial state. Nevertheless, we can = p(N +m)/2 q (N −m)/2 .
find out the probability of a particular outcome if we can [(N + m)/2]![(N − m)/2]!
assume that each outcome is equally likely2 . In such a case,
we obtain
1 Moments of a discrete distribution
P (X) = .
6
Before continuing with the random walk example, let us
This can, of course, be compared with the actual dice
recall a couple of concepts that characterize a given discrete
throws. Indeed, if we notice a deviation from this assump-
distribution (later, we will generalize these for continuous
tion of equally probable outcomes, we immediately suspect
distributions).
that the dice is crooked!
The mean of an arbitrary (normalized) distribution P (u)
2.2 Random Walk is given by
Next, we will consider one-dimensional random walk as a M
X M
X ui Ω(ui )
simple example, grasping the relevant concepts in probabil- µ ≡ hui = ui P (ui ) = lim , (6)
Ω(Σ)→∞ Ω(Σ)
ity calculus needed in this course. Assume, that a particle i=1 i=1
1 Ensemble is the French word for group. We will discuss in a later
where ui are the M possible values the variable u can have.
chapter the reasons why this kind of treatment can be done.
2 This is in fact a postulate of so-called a priori probabilities that The latter equality allows the interpretation of the mean
lies in the foundation of statistical physics, and will be discussed more as the ensemble average. The mean has the familiar inter-
thoroughly in the following chapter! pretation due to the law of large numbers:
3
STATISTICAL PHYSICS 3
Introduction to Statistical Physics
Statistical Physics
Physics
67 pages · professionally formatted edition
STUDY · REVIEW · REFERENCE
, Introduction to Statistical Physics
Johar M. Ashfaque
1
STATISTICAL PHYSICS 1
, 1. Introduction scope of the theory and leaves many interesting properties
of nature unexplained.
All matter we see around us consists of vast number of
microscopic particles that are in constant motion, and in The idea that the macroscopic properties of matter are
interaction with one another. At any given instant of time, caused by the microscopic constituents was first introduced
it is possible in principle to find the state of this kind of in the second half of the 19th century by James Clerk
many-body system. The laws of physics are such that the Maxwell, Ludwig Boltzmann, Max Planck, Rudolf Clau-
future evolution of the state can then be solved from the sius, and J. Willard Gibbs. The theory was completed in
governing equations of motion. However in practise, the the first half of 20th century by the advent of quantum
number of the constituent particles is typically of the order mechanics that gives the correct description of atomic size
of the Avogadro’s number particles. By applying the probabilistic ideas to systems
in equilibrium, one can obtain all results of the thermody-
NA = 6 · 1023 , namics together with the ability to relate the macroscopic
parameters of the system to its microscopic constituents.
leading to stupendously large set of coupled equations.
This powerful method will be subject in the first half of
Even the storing of the initial state of these equations
this course. The description of systems not in equilibrium
would exceed in multitude the combined memory of all
is much more difficult task. One can still make some gen-
of the computers in the world. This is the so-called many-
eral statements of such systems which leads to statistical
body problem of physics.
physics of irreversible processes. This will be discussed
When the number of the constituent particles is im- briefly in the latter part of the course.
mense, it has turned out useful to use the tools of probabil-
Statistical physics can, in principle, be applied to
ity calculus to describe their emergent behaviour. In this
any state of matter: solids, liquids, gases, matter com-
course, we will not be interested in the details of individ-
posed of several phases/or several components, matter
ual particles, but in the relations between the macroscopic
under extreme conditions of temperature and pressure,
properties that result. In fact, we lack most of the informa-
matter in equilibrium with radiation, and so on. In
tion required to describe the internal state of the system,
addition, the concepts and methods of statistical physics
and therefore our approach has to be probabilistic in na-
have turned out useful in many other fields of science:
ture. This is the starting point of statistical physics.
chemistry, study of dynamical systems, communications,
The equations governing the macroscopic properties (e.g. bioinformatics, complexity, and even stock markets.
energy, pressure, temperature) of a many-body system are
statistical averages of those of the constituent particles.
Therefore, it is implicitly assumed that the averaged quan-
tities experience statistical deviations.
For example, the familiar equation of state of the ideal “If a million monkeys typed ten hours a day, it is
gas extremely unlikely that their output would exactly equal
P V = nRT all the books of the richest libraries of the world; and yet,
relates the average pressure P and average volume V to in comparison, it is even more unlikely that the laws of
the average temperature T . Quite generally, the deviations statistical mechanics would ever be violated, even briefly.”
from the√average in a many-body system with N particles
go as 1/ N . The astounding number of particles, which - Emile Borel, 1913
makes the calculations of the macroscopic properties from
first principles practically impossible, leads to such an ac-
curacy of the statistical results that they can be taken as
exact physical laws.
Motivation
The systematic studies of macroscopic systems started
from the phenomenological studies in the 18th and 19th
centuries. These were done by the likes of James Watt,
William Rankine, Rudolf Clausius, Sadi Carnot, and
William Thomson. The resulted theory of thermodynam-
ics is concerned on heat and its relation to other forms of
energy and work. The strength of thermodynamics is in
its generality, emerging from the fact that the theory does
not depend on any detailed assumptions on the microscopic
properties of the system. This is also the main weakness
of thermodynamics, since only relatively few statements
can be made on such general grounds. This restricts the
2
STATISTICAL PHYSICS 2
, 2. Background performs successive steps, each of which has a length l and
is taken on a random direction. The successive steps are
2.1 Ensemble and Probability statistically independent, so we can denote
The laws of Physics are deterministic in a sense that p = probability of a right step
given the state of a system at one time one can calculate
the state at all later times, by using the time-dependent q = 1 − p = probability of a left step
Schrödinger equation or classical mechanics. Then, the
One obtains a great insight by considering the number p as
outcome of an experiment on the system should be deter-
the fraction of right steps in the ensemble of mental copies
mined completely by this final state. Even though this can
of single steps.
be done in principle, it is practically impossible to deter-
mine an initial state of a system with NA particles, let The random walk of N individual steps is formed by tak-
alone to calculate its time evolution. ing a subset, i.e. a sample, from the ensemble and calcu-
lating the sum of its members. After N steps, the particle
Statistical physics circumvents this problem by consid-
lies at
ering a (infinitely) large set of similar experiments, instead
x = ml,
of a single one. This kind of set of mental copies of similar
systems is called an ensemble 1 . where
Let then S denote a general system and consider an ex- m = nr − nl (2)
periment on the system with a general outcome X. We and nr (nl ) denotes the number of steps to right (left).
denote with Σ the ensemble of mental copies of S. Now, Naturally,
the probability of observing the outcome X is given by −N ≤ m ≤ N
Ω(X) and
P (X) = lim , (1)
Ω(Σ)→∞ Ω(Σ) N = nr + nl . (3)
where Ω(Σ) is the number of systems in the ensemble and
Ω(X) is the number of systems exhibiting the outcome X. By taking into account the number of different ways tak-
This is the scientific definition of probability. ing nr steps out of N , we arrive at the probability
N ! nr n l
Example: Throwing a dice WN (nr ) = p q (4)
nr !nl !
One could in principle determine the initial position and
of taking nr steps to the right. Probability function (4) is
orientation of a dice. Also, one could figure out exactly how
referred to as the binomial distribution since it resembles
the dice is thrown out of hand. By using these as initial
the terms in the binomial expansion.
conditions, one could use the classical equations of motion
to determine exactly which of the six numbers turns up. By using Eqs. (2) and (3), one can show (exercise) that
Practically, this is impossible. the probability of finding a particle at position m after N
steps is
Instead, we usually consider a large group of (nearly)
similarly thrown dices. The result of a single throw is PN (m) = WN (nr )
not anymore deterministic, since we do not have the ex-
N! (5)
act knowledge of the initial state. Nevertheless, we can = p(N +m)/2 q (N −m)/2 .
find out the probability of a particular outcome if we can [(N + m)/2]![(N − m)/2]!
assume that each outcome is equally likely2 . In such a case,
we obtain
1 Moments of a discrete distribution
P (X) = .
6
Before continuing with the random walk example, let us
This can, of course, be compared with the actual dice
recall a couple of concepts that characterize a given discrete
throws. Indeed, if we notice a deviation from this assump-
distribution (later, we will generalize these for continuous
tion of equally probable outcomes, we immediately suspect
distributions).
that the dice is crooked!
The mean of an arbitrary (normalized) distribution P (u)
2.2 Random Walk is given by
Next, we will consider one-dimensional random walk as a M
X M
X ui Ω(ui )
simple example, grasping the relevant concepts in probabil- µ ≡ hui = ui P (ui ) = lim , (6)
Ω(Σ)→∞ Ω(Σ)
ity calculus needed in this course. Assume, that a particle i=1 i=1
1 Ensemble is the French word for group. We will discuss in a later
where ui are the M possible values the variable u can have.
chapter the reasons why this kind of treatment can be done.
2 This is in fact a postulate of so-called a priori probabilities that The latter equality allows the interpretation of the mean
lies in the foundation of statistical physics, and will be discussed more as the ensemble average. The mean has the familiar inter-
thoroughly in the following chapter! pretation due to the law of large numbers:
3
STATISTICAL PHYSICS 3
