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  . 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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Statistical Mechanics 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  states, the probability for the isolated system in equilibrium to be in any one state is
P s

1

.
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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Statistical Mechanics Z. Suo 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 configuration A is
P
A



A .
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  states in which no molecule is in the top half of the
0
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Statistical Mechanics Z. Suo glass, Configuration 1 consists of  states in which 1 molecule is in the top half of the glass, and
1 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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Statistical Mechanics Z. Suo 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

N . Since all 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 probabilit y for N particles in volume V/10

/
N

V
V
10

N

10
N .
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. a set of
 tot
An isolated system has
 states. tot

 
1

 
2

 
3
The set of states is dissected into a family of configurations. Every state in a given configuration has a give value of Y.
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
Y i
, the system can be in a specific set of states; the number of states in this set is denoted by 
  i
. As the system switches from one state to another, the value of the variable Y fluctuates among a list of values,
Y
1
, Y
2
,...
Y i
,...
The total number of states  of the system is tot
 tot
 
 
1
 
 
2

...
 
  i

...
The sum is taken over all values of the variable.
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Statistical Mechanics Z. Suo
According to the fundamental postulate, the probability for the variable to take a specific value
Y j
is
P
    j
Y tot
.
The mean of the variable Y is
Y

P
 
1
Y
1

P
 
2
Y
2

...
The variance of Y is
Var
  
Y i

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 
 
states. According to the fundamental postulate, the most probable number of molecules in the gas maximizes the function 
 
. 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 
 
, 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


N
 dN

 
 

 

N dN .
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Statistical Mechanics Z. Suo
The more probable direction of change will increase the number of the states. Thus, if
 
/

N

0
, the gas will more probably gain molecules. If  
/

N

0
, the gas will more probably lose molecules.
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