Statistical Mechanics
Z. Suo
Isolated Systems
An isolated system. A system is a part of the world. For example, a glass of wine is a
system, 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.
A system is said to be an isolated system if it 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 block heat exchange, 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. Dose the magnetic field of the earth affect the wine?
If it does, we block it also.
Of course, nothing is perfectly isolated in reality. 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.
States of an isolated system. An isolated system has 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. The hydrogen atom interacts with the rest of the world, for example, by absorbing
photons. When the motion of the proton is neglected, the hydrogen atom has two states at the
ground energy level, six states at the second energy level, and so on. Thus, when the hydrogen
atom is isolated from the rest of the world, and has a fixed energy level, it can still be in several
states. An isolated system like a glass of wine has a great many states. We denote the number of
states of an isolated system by  .
An isolated system in equilibrium. An isolated system is not static: it constantly
switches from one state to another. For example, 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. The system can now
access some new states when molecules move around. The process takes time. Given enough
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time, the probability for an isolated system to be in any accessible state becomes independent of
time: the isolated system is said to be in equilibrium.
The fundamental postulate. Will a system be more probable in one state than another?
The fundamental postulate states that an isolated system in equilibrium is equally probable to be
in any one of its states. Thus, an isolated system in equilibrium behaves like a fair die. The
isolated system switches from one state to another, just as a madman rolls the die perpetually.
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.
For an isolated system with  states, the probability for the isolated system in
equilibrium to be in any one state is
Ps 
1
.

Configurations of an isolated system. A subset of the states of an isolated system is
called a configuration, a conformation or a macrostate of the system. For example, consider a
system consisting of a short RNA molecule in 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 conformation
of 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 may be differentiated by
biophysical methods. By contrast, the individual states may be too numerous to interest us.
For an isolated system with a total of  states, and a configuration A consisting of  A
states, the probability of the isolated system to be in macrostate A is
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PA 
A
.

Thus, once we accept the fundamental postulate, the statistical mechanics reduces to an art to
identify useful configurations, and then count the number of sates that constitute each
configuration.
Irreversibility. A configuration is more probable if it contributes more states to the
system. Now consider a half glass of wine, isolated from the rest of the world. Inside the glass is
a mixture of a liquid and a gas. Of all water molecules in the glass, some are in the liquid, and
others in the gas. We identify a configuration by a specific partition of water molecules in the
gas. Let us say we first seal the liquid, and evacuate the empty half of the glass. Then we open
the seal, and allow molecules to evaporate from the wine to fill the empty half of the glass with a
gas.
The initial partition does not contribute the largest number of states. After some time,
some other partitions will contribute more states and therefore be more probable than the initial
partition.
The partition of water has a small probability to go back to the initial value.
Consequently, the process from the initial to the equilibrium partition appears to be irreversible.
For a large system, the numbers of states associated with different configurations are very
different. Consequently, we replace the above statement of probability with a statement of
certainty: the system will always change toward a configuration that increases the number of
states.
Ink particles. 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. Why do the ink particles disperse? At the beginning, all
the ink particles are in a small volume in the wine. As time proceeds, each ink particle is free to
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Statistical Mechanics
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explore 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
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,
probabilit y for N particles in volume V
VN

 10 N .
N
probabilit y for N particles in volume V/10 V / 10
This ratio is huge if we have more than a few ink particles, a fact that explains why the ink
particles prefer dispersion to localization.
Specify a family of configurations by using a variable. We often specify a family of
configurations by using a variable. For example, the number of water molecules in the half glass
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of wine specifies a family of configurations. When a variable Y is held at a specific value Yi , the
system can be in a specific set of states; the number of states in this set is denoted by Yi  . As
the system switches from one state to another, the value of the variable Y fluctuates among a list
of values, Y1, Y2 ,...Yi ,... The total number of states tot of the system is
tot  Y1   Y2   ...  Yi   ...
The sum is taken over all values of the variable.
According to the fundamental postulate, the probability for the variable to take a specific
value Y j is
PY j  
Y j 
tot
.
The mean of the variable Y is
Y  PY1 Y1  PY2 Y2  ...
The variance of Y is
Var Y    Yi  Y

2
These sums are taken over all values of the internal variable.
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