Statistical Thermodynamics
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Pressure
Work done by a pressure applied to a system. Subject the surface of a system to a
pressure. The pressure is taken to be uniform from one part of the surface to another part, but the
magnitude of the pressure, p, may vary with time. We want to calculate the work done by the
pressure. By definition, the work done by a force f over a small distance dr is fdr . The action
of the pressure on a small area a gives a force pa. When this small area moves a distance in the
direction dr along the normal of the area, the force does work padr . Because the pressure is
taken to be uniform over the surface of the system, the work by the pressure to the system is
 pdV , where dV is the change in the volume of the system, and the negative sign conforms to
the convention: the work due the pressure is positive when the volume of the system decreases.
Enthalpy. Consider a specific situation, in which a weight applies the pressure to a
system through an incompressible fluid. The system can change its volume V. The potential
energy of the weight is the weight times the height, which is the same as pV. We regard the
weight and the system as a composite, with the total energy
H  U  pV .
When the weight and the system is not in equilibrium, p can take any value. When the weight is
in equilibrium with the system, p must take a value specific to the system. Consequently, in
equilibrium, the quantity H is specific to the system, and is
known as the potential energy of the system in mechanics, and as
weight
the enthalpy of the system in thermodynamics.
Incompressible fluid
A system that can change both energy and volume.
Now we open the system in two ways: its energy U and its
volume V can vary independently. We block all other modes of
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interaction. When both the energy and the volume are held constant, the system is an isolated
system, and the number of states of the system is  . The function U,V  characterizes a
family of isolated system.
When V is held constant, and U is allowed to vary, the system is in thermal contact with
its environment, as discussed before. We have already defined the temperature T by
1  log U ,V 

.
T
U
When the energy and the volume vary by small amounts, dU and dV, the function
log U,V  varies by
d log  
 log U ,V 
 log U ,V 
dU 
dV .
U
V
We have interpreted one differential coefficient by the temperature, and would like to interpret
the other differential coefficient.
To do so, we make the composite of the weight and the system an isolated system, with a
fixed total energy H . Here we apply a constant weight, which gives a constant pressure. We
now regard the volume of the system, V, as the internal variable of the composite. We ask, when
the volume is allowed to vary, what is the volume as the weight equilibrates with the composite?
When the volume changes by dV, because the total energy of the composite H and the
pressure p are both constant, the energy of the system changes by dU   pdV . The weight
contributes no additional states to the composite, so that the number of states of the composite is
still U,V  . When the volume is allowed to vary, the isolated system reaches equilibrium
when the number of states U,V  reaches maximum. Substituting d log   0 and
dU   pdV into the differential expression of log U,V  , we obtain that
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p T
 log U ,V 
.
V
That is, when the weight and the system equilibrate, the pressure due to the weight must be
related to the quantities characteristic of the system.
We have now interpreted both differential coefficients of the function log U,V  .
When the energy and the volume vary by small amounts, dU and dV, the function log U,V 
varies by
d log  
1
p
dU  dV .
T
T
Ideal gas. A container of volume V contains N molecules. The kinetic energy of the
molecules is so large that intermolecular interaction is negligible. Furthermore, the distance
between the molecules is so large that the probability of finding a molecule is independent of the
location in the container, and of the presence of other molecules. The total number of states of
the system is proportional to the total number of ways in which the N molecules can be
distributed. The latter equals the product of the numbers of ways in which the individual
molecules can be independently distributed. With N and U fixed, each of these numbers will be
proportional to V. The number of states is proportional to the Nth power of V:
U ,V   V N .
The proportional factor will depend on U and N, but not on V.
Inserting this expression for the number of states into the expression of the pressure,
p T
 log 
,
V
we obtain that
p  TN / V .
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This is the familiar ideal gas law. A bag of air acts like a spring. The volume decreases when
the pressure increases, and recovers when the pressure drops. This elasticity clearly does not
result from distortion of bonds in the molecules, but from the fact that the number of states
increases with the volume. Such elasticity is known as entropic elasticity.
Osmosis. A bag contains a liquid of volume V, with N particles dispersed in the liquid.
The particles can be of any size. When the particles are molecules, we call them solutes. When
the particles are somewhat larger, say from 10 nm to 10  m, we call them colloids. The bag is
immersed in a reservoir of the same liquid but without any such
Liquid
particles. The liquid is incompressible, but we can change the
volume of the liquid inside the bag by allowing the molecules
of the liquid to permeate through the skin of the bag. The
particles in the bag, however, cannot permeate through the skin.
Such a skin is semi-permeable.
The physics of this situation is analogous to the ideal gas, provided that the concentration
of the particles is dilute. Every particle is free to explore the entire volume in the bag. The
number of states of the N particles in volume V scales as   V N . The liquid molecules
permeate through the skin to drive the composite system (the bag and the reservoir) to reach
equilibrium. Consequently, V is a variable. Recall once again the defining equation of the
pressure,
p T
 log 
.
V
Inserting the expression   V N , we obtain that
p  TN / V .
This pressure is known as the osmotic pressure.
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In equilibrium, the osmotic pressure can be balanced in several ways. For example, for a
spherical bag, the membrane tension can balance the osmosis pressure. One can also disperse
particles in the reservoir, and make sure that the particles do not permeate into the bag. The
pressures in the reservoir balances that in bag provided the concentrations of the particles are
equal. As yet another example, we place a rigid, semi-permeable wall in the liquid, with the
particles on one side, but not the other. Water is on both sides of the wall, but alcohol is only on
one side. The molecules of the liquid can diffuse across the wall, but the particles cannot. For
the particles to explore more volume, the liquid molecules have to diffuse into the side where
particles are.
If this experiment were carried out the in the zero-gravity environment, the
infusion would continue until the pure liquid is depleted. In the gravitation field, the infusion
stops when the pressure in the solution balances the tendency of the infusion.
The internal energy U S ,V  . Usually,  is an increasing function of U, when V is held
constant. That is, everything else being fixed, the more energy a system has, the more number of
states. Thus, we can invert the function U,V , and consider the function U ,V  . We can
vary the energy of a system in two ways, by varying the number of states or by varying the
volume. We have called the function S U ,V   log U ,V  the entropy of the system. Write the
function as U S ,V  . Take partial differentiation, and we obtain that
dU 
U S ,V 
U S ,V 
dS 
dV .
S
V
The second term is the work done to the system when the system held the number of states
constant. We can rewrite the above as
dU  TdS  pdV .
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The first term represents heat added to the system, and the second term represents the work done
by the external pressure on the system.
Equilibrium upon lifting an internal constraint. Let Y be an internal variable, and a
family of isolated systems be characterized by a function S U ,V , Y  .
Recall a consequence of
the fundamental postulate: When U and V are held constant, and Y is free to vary, a value of
Y is more probable if it gives the system the larger entropy. Of course, the two functions,
S U ,V , Y  and U S ,V , Y  , characterize the same family of isolated systems. Thus, When S and
V are held constant values and Y is free to vary, a value of Y is more probable if it gives the
system smaller energy. For a large system, we usually replace the above statement of probability
with a statement of certainty.
The enthalpy H S , p  . We can also register the interaction between a system and the
rests of the world using other independent variables. The change of variable can be done by a
procedure known as the Legendre transformation.
To illustrate the procedure, define a function
by
H  U  pV .
The function has the unit of energy, and is known as the enthalpy of the system. For small
changes in variables, we obtain that
dH  dU  d  pV  .
Recall the expression dU  TdS  pdV , and an identity in calculus d  pV   pdV  Vdp . We
obtain that
dH  TdS  Vdp .
The coefficients in the differential form can be deified by partial derivatives of the function
H S , p  :
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T
H S , p 
H S , p 
.
,V
S
p
In this example, the Legendre transformation replaces V by p as an independent variable.
We can think the above change of variable in physical terms. Imaging that the system is
sealed in a cylinder with a piston, which is frictionless. A weight is placed on the piston to
balance the pressure inside the system. Now we can regard the system and the weight as a
composite. The energy of the weight is pV. The total energy of the composite is the sum of the
energy of the system and that of the weight, namely, U  pV . We can apply the principle of
energy to the composite. When S, p  are held constant values and Y is free to vary, a value of
Y is more probable if it gives the system smaller enthalpy.
The Helmholtz free energy F T ,V  . The Legendre transformation can be applied to
other variables. As another example, define the enthalpy by
F  U  ST .
Its differential is
dF   SdT  pdV .
Consequently, we can regard F as a function of T,V  , and the coefficients in the above
differential form are the partial derivatives of the function F T ,V  .
In this example, the
Legendre transformation replaces S by T as an independent variable.
We can now paraphrase the equilibrium condition as follows. When a system is held at a
constant temperature, upon lifting a constraint internal to the system, after a long time, the more
probably value of the internal variable has a smaller value of the Helmholtz free energy.
A small system in thermal equilibrium with a large system. If you find the above
change of variable too abstract, you can always go back to the fundamental postulate and count
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the number of states. To prescribe a temperature to a system, we can bring the system to thermal
contact with a much larger system. The two systems can only exchange energy, but all other
modes of interaction are blocked. The large system acts as a reservoir, from which the small
system can draw energy. Because the reservoir is a large system with many degrees of freedom,
its number of states,  R , is a smooth function of its energy U R , and its temperature, TR , is
defined by
 log  R
1

.
U R
TR
We make the composite of the reservoir and the given system an isolated system, with a
fixed total energy U tot .
When the system has energy U, the reservoir has the remainder,
U R  U tot  U . The energy of the system is small compared to the total energy, U  U tot , so
that the temperature of the reservoir, TR , remains constant as the system draws energy from the
reservoir. From the above definition of the temperature, log R is linear in U R , namely,
log  R U tot  U   log  R U tot  
U
.
TR
Take exponential of both sides of the equation, and we obtain that
R Utot  U   R Utot exp  U / TR  .
Note that  R U tot  is the number of states of the reservoir at energy U tot . Upon losing energy U
to the system, the reservoir reduces its number of states by a factor
exp  U / TR  ,
known as the Boltzmann factor. We assume that this energy exchange is slow enough so that the
reservoir has enough time to access all the R U tot  U  states. That is, after giving energy U to
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the system, the reservoir itself has relaxed to an isolated system in equilibrium, with no internal
variables.
By contrast, we allow the small system to have an internal variable Y. When held at
constant U, Y  , the small system is also an isolated system, whose number of states is a function
U, Y  . If the small system has just a few degrees of freedom, U, Y  may not be a smooth
function, so that the temperature of the small system may not be well defined. In the following
discussion we will not invoke the temperature of the system, unless we are sure that the system
also has many degrees of freedom.
When the small system is held at constant U, Y  , the number of states of the composite,
com , is the product of the number of the states of the reservoir and that of the system:
 U
com   R U tot exp   U , Y  .
 TR 
At a fixed Y, the energy of the system, U, is an internal variable of the composite. When U
increases, the system has more states, but the reservoir has fewer states. Once the composite
becomes an isolated system in equilibrium, the fundamental postulate dictates that the most
probable value of U maximizes com .
By definition, U, Y  relates to the entropy S U ,Y  of the system as
U ,Y   expS U ,Y  .
Thus, we can rewrite com as
 U  TR S U , Y 
com   R U tot exp 
.
TR


In the above expression, only the following quantity varies with U, Y  :
U  TR S U ,Y  .
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When the system is in contact with the reservoir held at a constant temperature TR , but U, Y  of
the system can vary, a pair of U, Y  is more probable if it gives a smaller value of
U  TR S U ,Y  , and the most probable values of U, Y  minimizes the quantity U  TR S U ,Y  .
Note that the quantity U  TR S U ,Y  looks like the Helmholtz free energy of the system, except
that the TR is not the temperature of the system, but is the temperature of the reservoir.
When the small system has many degrees of freedom, its temperature is defined by
1 S

.
T U
Consider the following scenario. The system has been in contact with the reservoir long enough
to equalize the temperature. However, there is still a constraint internal to the system. Upon
lifting this constraint, as the internal variable Y changes, the number of states of the composite
can further increase. Now we can replace the temperature of the reservoir in the quantity
U  TR S U ,Y  by the temperature of the system, T , and call the quantity
F  U  ST
by its proper name the Helmholtz free energy of the system. We simply say that the system is
held at a constant temperature T. The system is not isolated, and opens in a single way: its
energy can change. When a system is held at a constant temperature, upon lifting a constraint
internal to the system, after a long time, the more probably value of the internal variable has a
smaller value of the Helmholtz free energy.
Let us look the above statement at the limits of low and high temperatures. At a low
temperature, the free energy is dominated by the energy, and the more probable Y gives lower
energy to the system. At a high temperature, the free energy is dominated by the entropy term,
and the more probable Y gives higher entropy to the system. At an intermediate temperature, the
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more probable Y reduces the free energy, which compromises the tendency to reduce the energy
and that to increase the entropy. At room temperature, a hard material, such as a metal or a
ceramic, is largely governed by energy reduction, while a soft material, such as a rubber or a
colloidal suspension, is largely governed by entropy increase.
Gibbs free energy GT , p  . As yet another example of the Legendre transformation,
define the Gibbs free energy by
G  U  TS  pV .
Its differential form is
dG   SdT  Vdp .
The coefficients in the differential form are defined by the partial derivatives of the function
GT , p  .
In physical terms, this change of variables means that the system of brought in contact
with a reservoir, such that the system can change both energy and volume, but not particles. One
can similarly state the equilibrium condition as a configuration that minimizes G.
The Maxwell relations. Recall an identity in calculus: given a differential function
f x, y  , the partial derivatives are indifferent to the order by which they are taken. Thus,
 2 f x, y   2 f x, y 
.

xy
yx
Applying this identity to the first two variables in the function U S ,V  , we obtain that
 2U S ,V   2U S ,V 
,

SV
VS
or
T S ,V 
pS ,V 

.
V
S
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The procedure can be applied to any other functions and any other pairs of the variables.
For example, applying the procedure to the function F T ,V  and variables T and V, we obtain
that
S V , T  p V , T 

.
V
T
Thermoelasticity. All materials change length with temperature, and exchange heat with
the environment when stretched. To study these phenomena, we use temperature T and the
length L as independent variables, and use the Helmholtz free energy F T , L . When the
temperature and the length change by a small amount, the free energy changes by
dF   SdT  fdL .
The entropy is given by one partial differentiation:
S 
F T , L 
,
T
and the force is given by another partial differentiation:
f 
F T , L 
.
L
The Maxwell equation is

S f

.
L T
For a small change in the temperature and length, the entropy and the force change by
dS 
c
dT  kadL ,
T
df  kadT  kdL .
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Here we have introduced the heat capacitance c and stiffness k. The Maxwell relation relates the
coefficients of the cross terms, which we have written as ka. In general, the three independent
coefficients, c, k and a are functions of T, L .
If the temperature and the length do not change too much, the coefficients can be treated
as constants. When force is zero, the length changes with temperature linearly:
L0  aT  b .
Force is linearly proportional to the change in length (Hooke’s law):
f T , L   k L  L0  .
where we neglect the temperature dependence of the stiffness.
Elastic dielectric. All materials contain electrons and protons.
In a dielectric, these
charged particles form bonds, and move relative to one another by short distances in response to
a voltage or a force. That is, all dielectrics are deformable. The notion of a rigid dielectric is as
fictitious as that of a rigid body: they are idealizations useful for some purposes, but misleading
for others.
Work. The following figure illustrates a system of insulators and conductors, loaded by a
field of weights and batteries, of which only one of each is drawn. All batteries are connected to
a common ground. We can measure the displacement l of the weight, and the amount of
charge Q pumped by the battery from the ground to the electrode. There might be other
weights dropping or rising and other batteries pumping charge from or to the ground, but the
work done by this particular weight is Pl , and the work done by this particular battery is Q .
If we regard work, displacement and charge as primitive, measurable quantities, the above
statements of work define the force P supplied by the weight, and the voltage  supplied by the
battery. The force is said to be the work conjugate to the displacement, and the voltage the work
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conjugate to the charge. We will use the word weight as shorthand for all mechanisms (including
inertia) that do work through displacements, and the word battery as shorthand for all
mechanisms that do work through flows of charge. We will neglect the effects of magnetism and
electromagnetic radiation.
electrode
dielectric
Q

ground
P
l
Electromechanical coupling. Now imagine that the weight and battery are adjustable, so
that the force P and the voltage  can vary. When the displacement is held constant, a change
in the charge may cause the force to change. When the charge is held constant, a change in the
displacement may cause the voltage to change. These electromechanical coupling effects are
universal to all dielectrics, because all dielectrics have electrons and protons, and the charged
particles can move relative to one another.
Conservative system.
The two electromechanical coupling effects are linked for
conservative systems. A conservative system is one for which, under isothermal conditions, the
work done by the weight and the battery is fully stored as the Helmholtz free energy of the
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system. That is, associated with small changes l and Q , the free energy of the system, F,
changes by
F  Pl  Q .
To this equation we should add the work done by all other weights and batteries. For simplicity,
however, here we assume that only one weight and one battery do work. This may be achieved
by removing all other weights and batteries, and making sure that every other part in the system
other than the particular electrode is either grounded or charge neutral. The temperature is held
constant, and we do not list it as a variable.
These idealizations ensure that the free energy of the system is a function of two
variables, F l , Q . We only need to measure the difference in F, l and Q between the current
state and a reference state. For the conservative system, the force and the voltage are partial
derivatives:
P
F l , Q 
,
l

F l , Q 
.
Q
Associated with small changes l and Q , the force and the voltage change by
P 
 2 F l , Q 
 2 F l , Q 

l

Q ,
l 2
lQ
 
 2 F l , Q 
 2 F l , Q 
l 
Q .
lQ
Q 2
We may call  2 F l , Q  / l 2 the mechanical tangent stiffness of the system, and  2 F l , Q  / Q 2
the electrical tangent stiffness of the system. The two electromechanical coupling effects are
both characterized by the same cross derivative, namely,
Pl , Q   2 F l , Q  l , Q 
.


Q
lQ
l
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Consequently, for a conservative system, the two electromechanical coupling effects reciprocate.
Vacuum. As an illustration, consider the parallel-plate capacitor, loaded by both the
voltage and the weight. The two electrodes are separated by a vacuum. The separation between
the two electrodes l may vary, but the area of either electrode remains to be A. Recall the
elementary fact:
U l , Q  
lQ 2
,
2 0 A
where  0 is the permittivity of vacuum, so that
P
U l , Q 
Q2
.

l
2 A 0
This force is due to the attraction of the opposite charges on the two electrodes, and is balanced
by the weight.
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