Statistical Mechanics
Z. Suo
Isolated Systems
Our world has the following empirical facts:
•
An isolated system has a set of quantum states.
•
The isolated system flips ceaselessly from one quantum state to another.
•
After being isolated for a long time, the isolated system keeps flipping from one quantum
state to another, and has the equal probability to be in any one of the quantum states.
Thus, the thermodynamics of an isolated system is identical to the theory of probability of rolling
a fair die. These notes remind you what an isolated system is, and how it relates to happenings
in our world. Also reviewed are rudimentary ideas of probability, and their translation to the
thermodynamics of an isolated system.
A system. We are egocentric: we call everything detectable by us the world. We regard
as we please any part of the world a system. A glass of wine, for example, is a system. This
system is composed of water, alcohol, and other molecules. Do we include the glass as a part of
the system? Maybe. The decision is ours. We can regard any part of the world as a system.
An isolated system. A system is said to be isolated if the system does not interact with
the rest of the world. To make a glass of wine an isolated system, we seal the glass to prevent
molecules from escaping, place the glass in a thermos to prevent energy from escaping, and
make the seal so stiff that the external pressure cannot do work to the wine. We are alert to any
other modes of interaction between the wine and the rest of the world. Does the magnetic field
of the earth affect the wine? If it does, we will shield the glass of wine from the magnetic field of
the earth also.
Of course, nothing is perfectly isolated. Like any idealization, the isolated system is a
useful approximation of the reality, so long as the interaction between the system and the rest of
the world negligibly affects a phenomenon of interest to us. We will use the notion of the
isolated system to build the entire thermodynamics, including the thermodynamics of open
systems. But for the time being, let us focus on isolated systems.
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Statistical Mechanics
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An isolated system has a set of quantum states. An isolated system can be in a set
of stationary quantum states, or states for brevity. A hydrogen atom, for example, is a system
composed of a proton and an electron. When the motion of the proton is neglected, the state of
the hydrogen atom is characterized by the electron cloud. The hydrogen atom interacts with the
rest of the world, for example, by absorbing photons, so that the energy of the electron can
change. Even when the hydrogen atom is isolated from the rest of the world, the isolated system
can still be in multiple states. The hydrogen atom has two states if the atom is isolated at the
ground energy level, six states if the atom is isolated at the second energy level, and so on.
An isolated system like a glass of wine has a huge number of states. The glass of wine as
a whole is a quantum system. A state of this system describes collectively the cloud of all
electrons and positions of all protons.
An isolated system ceaselessly flips from one quantum state to another. An
isolated system is not static: it constantly switches from one state to another. Thus, an isolated
system behaves like a die of a large number of faces, and a madman rolls the die perpetually. For
example, in the glass of wine, molecules ceaselessly move.
An isolated system in equilibrium. Imagine that we add a drop of water into the
glass of wine, and then isolate the new system from the rest of the world. In the beginning, the
probability of the isolated system to be in a state changes with time. Given enough time,
however, the probability for the isolated system to be in any state becomes independent of time:
the isolated system is said to be in equilibrium.
The fundamental postulate. Here are two basic facts of our world:
1. an isolated system has a set of quantum states, and
2. the isolated system ceaselessly flips from one state to another.
Will the isolated system be more probable in one quantum state than another? The fundamental
postulate states that a system isolated for a long time has the same probability to be in any one
of the quantum states.
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Thus, an isolated system in equilibrium behaves like a fair die of a large number of faces.
Every state of the isolated system in equilibrium is equally probable, just as every face of a fair
die is equally probable. The fundamental postulate cannot be proved from more elementary
facts, but its predictions have been confirmed without exception by empirical observations. We
will regard the fundamental postulate as a basic empirical fact. The entire thermodynamics rests
on the fundamental postulate.
We next describe several elementary consequences of the
fundamental postulate.
The number of states of an isolated system. Each isolated system has a set of
states, the total number of which is denoted by Ω . The number of states is a fundamental
property of an isolated system. In principle, for a given isolated system, its number of states
may be calculated using quantum mechanics. In practice, however, such a calculation can only
be done for a few idealized systems. We hasten to note that the number of states of an isolated
system can be readily measured by experiments, as we will describe later.
An isolated system behaves like a fair die. The theory of probability of a fair die can be
translated into the thermodynamics of an isolated system. For an isolated system with a total of
Ω states, the probability for the isolated system in equilibrium to be in any one state is
1
.
Ω
A macrostate of an isolated system. A subset of the states of an isolated system is
called a macrostate of the isolated system. Depending on the type of the system, we may also
use other words to designate a macrostate of the system. For example, we may be interested in a
conformation of a molecule, or a thermodynamic state of a glass of wine, or a configuration of
ink particles in a liquid.
(In the theory of probability, rolling a die is called an experiment, each state of the die an
outcome, and a subset of states an event.)
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Consider a specific system: a short RNA molecule in a large quantity of a liquid. Let us
say that the RNA molecule can be in two conformations: chains or loops. Each conformation is a
gross description, consisting of many states. For example, both a straight chain and a wiggled
chain belong to the same conformation, the chains. Even when the shape of the RNA molecule is
fixed, the molecules in the surrounding liquid can take many alterative configurations. The two
conformations of the molecule may be differentiated by biophysical methods. By contrast, the
individual states may be too numerous to interest us.
The probability of an isolated system to be in a macrostate. Consider an
isolated system with a total of Ω states. Then consider a macrostate A of the isolated system.
The macrostate consists of Ω A states of the isolated system. The fundamental postulate requires
that the probability of the isolated system to be in macrostate A be
ΩA
.
Ω
Thus, once we accept the fundamental postulate, the entire thermodynamics reduces to an art to
identify useful macrostates, and then count the number of sates that constitute each macrostate.
Irreversible process in an isolated system. Now consider a half glass of wine. We
seal the bottom half of the glass, evacuate the top half of the glass, and isolate the whole glass
from the rest of the world. Then we remove the seal, and allow molecules to escape from the
liquid to fill the top half of the glass with a gas. Our experience indicates that the process of
evaporation is spontaneous, but the molecules in the gas will not spontaneously all go back to
the liquid. That is, the evaporation of a liquid into a vacuum is an irreversible process in an
isolated system. What causes this irreversibility?
We can explain this irreversibility in terms of the fundamental postulate. The seal in the
middle of the glass provides a constraint internal to the isolated system. The act of removing the
seal lifts the constraint, making the number of molecules in the top half of the glass a variable.
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We label a macrostate of the isolated system by the number of molecules in the top half of the
glass. Thus,
•
Macrostate 0 consists of Ω0 states in which no molecule is in the top half of the glass,
•
Macrostate 1 consists of Ω1 states in which 1 molecule is in the top half of the glass,
and so on.
According to the fundamental postulate, a macrostate of an isolated system is more
probable if the macrostate consists of more states of the isolated system. When the constraint is
lifted, there will be other macrostates that may consist of more states of the isolated system.
Consequently, the process from the initial to the final macrostate appears to be irreversible.
Dispersion of a drop of ink in a glass of wine. To have some feel for numbers,
consider a drop of ink in a glass of wine. The ink contains small solid particles (e.g., carbon
black) that give the color.
After some time, the ink particles disperse in the wine. Our
experience indicates that dispersion of ink in a liquid is an irreversible process: the ink particles
will spontaneously disperse in the liquid, but the dispersed ink particles will not spontaneous all
come back to form a drop of ink. What causes the irreversible dispersion?
At the beginning, all the ink particles are in a small volume in the wine.
As time
proceeds, each ink particle is free to move in the entire volume of the wine. A list of positions of
all ink particles defines a configuration of the system. For a given configuration, the system can
still be in many states. Consequently, each configuration corresponds to a large number of states
of the system. All configurations are equally probable. A configuration that all ink particles
localize in a small region in the glass is just as probable as a configuration that the ink particles
disperse in the entire glass. However, there are many more configurations that the ink particles
disperse in the entire glass than the configurations that the ink particles localize in a small
region. Consequently, dispersion is more likely than localization.
How many more likely? Let us make this idea quantitative. We view the wine and the
ink as a single system, and isolate the system from the rest of the world. Let V be the volume of
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the glass of wine, and N be the number of the ink particles. We have a dilute concentration of
the ink particles suspended in the wine. The interaction between the ink particles is negligible,
so that each particle is free to explore everywhere in the wine. Consequently, the number of
configurations of each ink particle is proportional to V. The number of configurations of the N
ink particles is proportional to V N . On the other hand, if the N particles localize in a small
region, say of volume V/10, the number of configurations is proportional to (V / 10 ) . Since all
N
configurations are equally likely, the probability to find the N ink particles in a given volume is
proportional to the number of configurations. Thus,
probability for N particles in volume V
VN
=
= 10 N .
probability for N particles in volume V/10 (V / 10 )N
This ratio is huge if we have more than a few ink particles, a fact that explains why the ink
particles much, much prefer dispersion to localization.
A function of state. Given an isolated system, and consider a function that maps a
state of the isolated system to a number. That is, the domain of the function is the set of all the
states of the isolated system. The range of the function is a real number, a quantity that interests
us. In the theory of probability, such a function of state is called a random variable. For
example, for the isolated system of a half glass of wine, we are interested in the number of
molecules in the top half of the glass, i.e., the number of molecules in the gas. This number is a
function of the state of the isolated system. Such a function is usually not a 1-1 function: many
states of the isolated system give the same number of molecules in the gas.
Dissect the whole set of states of an isolated system into a family of
configurations by using a function of state. Consider an isolated system, and a function
that maps each state of the isolated system to a real variable, Y. When the variable is held at a
specific value Yi , the isolated system can be in any one of a subset of states; the number of states
in this subset is denoted by Ω(Yi ) . As the system flips from one state to another, the value of the
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variable Y fluctuates among a list of values, Y1 , Y2 ,...Yi ,... The total number of states Ωtot of the
isolated system is
Ωtot = Ω(Y1 ) + Ω(Y2 ) + ... + Ω(Yi ) + ...
The sum is taken over all values of the variable.
Ω(Y1 )
Ω tot
Ω(Y2 )
An isolated system has
a set of Ωtot states.
Ω(Y3 )
The set of states is dissected into
a family of subsets. Each subset
is known as a configuration. All
states in a given configuration
have the same value of Y.
In contrast to using a natural language (English, Chinese, etc.) to describe
configurations, using a real variable to dissect the set of all states of an isolated system into a
family of configurations has an obvious advantage: the variable will allow us to use mathematics
more effectively. One game of thermodynamics is to identify a function of state for an isolated
system, and use the function to classify all the states of the isolated system into a family of
configurations.
Rudimentary ideas of probability. As noted above, the fundamental postulate turns
the thermodynamics of an isolated system into the theory of probability of a fair die. Let us list a
few more rudimentary ideas of probability. An isolated system has a total of Ωtot states. When
the isolated system is in any one of Ω(Y j ) states, the variable Y takes the value Y j The
probability for the variable to take a specific value Y j is designated by P (Y j ) . According to the
fundamental postulate,
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Z. Suo
P (Y j ) =
Ω(Y j )
Ωtot
.
The mean of the variable Y is
Y = P (Y1 )Y1 + P (Y2 )Y2 + ...
The variance of Y is
Var (Y ) =
∑ (Y − Y ) .
2
i
These sums are taken over all values of the variable Y.
Is there any difference between the thermodynamics of an isolated system
and the theory of probability of rolling a fair die? None. Except the means by which we
determine the probabilities of various configurations. The theory of probability seems to be
obsessed with determining the probabilities by using calculations of combinatorics. As we will
describe later, the thermodynamics of an isolated system mostly determines the probabilities by
using experiments with the isolated system.
We might as well regard thermodynamics as
experimental combinatorics.
Rudimentary ideas of calculus. Consider the half glass of wine again. When the
number of molecules in the top half of the glass is N , the isolated system has Ω( N ) states. For
most isolated systems, the number of molecules is so large that we may as well regard N as a
continuous variable. This simplification allows us to use calculus.
For example, if we know the function Ω( N ) , and know the number of molecules in the
gas N, we can ask if the gas will gain or lose molecules. For a small change in the number of
molecules in the gas, dN , the number of states of the isolated system changes by
Ω( N + dN ) − Ω( N ) =
∂Ω
dN .
∂N
Of the two directions of change, gaining and losing molecules for the gas, the more probable
direction of change increases the number of the states. This statement is a direct consequence of
the fundamental postulate, and is translated into the following conclusions.
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•
If ∂Ω / ∂N > 0 , the gas will more probably gain molecules.
•
If ∂Ω / ∂N < 0 , the gas will more probably lose molecules.
•
If ∂Ω / ∂N = 0 , the gas will be equally probable to gain or lose molecules.
From probability to (almost) certainty. According to the fundamental postulate,
the most probable number of molecules in the gas, N, maximizes the function Ω( N ) . The
isolated system of the half glass of wine is said to reach a thermodynamic state of equilibrium.
Or we say that the liquid equilibrates with the gas.
Any other partitions of molecules are also probable, but are less probable than the
equilibrium partition. Our experience indicates that, after a long time, the equilibrium partition
is much, much more probable than other partitions. We therefore often focus our attention on
the partition that maximizes the function, Ω( N ) , rather than be bothered with the probability of
less probable partitions. This is why a large part of thermodynamics is an endless game of
minimization.
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