Statistical Mechanics
Z. Suo
Isolated Systems
A system. We call everything that we can detect the world. Any part of the world may
be regarded as a system. 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.
An isolated 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, it can still be in multiple 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 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 highly
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Z. Suo
idealized systems. We hasten to remark that the number of states of an isolated system can be
readily measured by experiments, as we will describe later.
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. 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. 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 of a large
number of faces. 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 Ω states, the probability for the isolated system in
equilibrium to be in any one state is
Ps =
1
.
Ω
The entire thermodynamics rests on this fundamental postulate. We next give a few elementary
consequence of the fundamental postulate.
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
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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 Ω states, and a configuration A consisting of Ω A
states, the probability of the isolated system to be in configuration A is
PA =
ΩA
.
Ω
Thus, once we accept the fundamental postulate, thermodynamics reduces to an art to identify
useful configurations, and then count the number of sates that constitute each configuration.
Irreversibility. 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.
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.
We label a configuration of the isolated system by the number of molecules in the top half of the
glass. Thus, Configuration 0 consists of Ω0 states in which no molecule is in the top half of the
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glass, Configuration 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 configuration of an isolated system is more
probable if the configuration consists of more states of the isolated system. When the constraint
is lifted, there will be other configurations that may consist of more states of the isolated system.
Consequently, the process from the initial to the final configuration appears to be irreversible.
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
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
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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 .
N
probability 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.
Ω(Y1 )
Ω tot
Ω(Y2 )
Ω(Y3 )
The set of states is dissected
into
a
family
of
configurations. Every state
in a given configuration has
a give value of Y.
An isolated system has
a set of Ω tot states.
Dissect the set of states of an isolated system into a family of configurations by using
a variable. In the example of a half glass of wine, we identify a configuration by the number of
molecules in the top half of the glass. More generally, 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 Ω(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 Ωtot of the system is
Ωtot = Ω(Y1 ) + Ω(Y2 ) + ... + Ω(Yi ) + ...
The sum is taken over all values of the variable.
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According to the fundamental postulate, the probability for the variable to take a specific
value Y j is
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 ) = ∑ (Yi − Y
)
2
.
These sums are taken over all values of the variable Y.
In contrast to using a natural language (English, Chinese, etc.) to describe configurations,
using a 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 to turn
trivial observations into impressive (and useful) results.
As another example, let us 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. According to
the fundamental postulate, the most probable number of molecules in the gas maximizes the
function Ω( N ) . For most applications, the number of molecules are so large that we may as well
regard N as a continuous variable. If we know the function Ω( N ) , and know the current
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
change by
Ω( N + dN ) − Ω( N ) =
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∂Ω
dN .
∂N
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The more probable direction of change will increase the number of the states.
Thus, if
∂Ω / ∂N > 0 , the gas will more probably gain molecules. If ∂Ω / ∂N < 0 , the gas will more
probably lose molecules.
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