identi-…ed with a point in phase space. Let us now consider the di¤usion of a gas in an initial state (qi(t0); pi(t0)) from a smaller into a larger volume. If one is really able to reverse all momenta in the …nal state (qi(tf); pi(tf))and to prepare a state (qi(tf); pi(tf)), the process would in fact be reversed. From a statistical point of view, however, this is an event with an incredibly small probability. For there is only one point (microstate) in phase space which leads to an exact reversal of the process, namely (qi(tf); pi(tf)). The great major-ity of microstates belonging to a certain macrostate, however, lead under time reversal to states which cannot be distinguished macroscopically from the …nal state (i.e., the equilibrium or Maxwell- Boltzmann distribution). The funda-mental assumption of statistical mechanics now is that all microstates which have the same total energy can be found with equal probability. This, however, means that the microstate (qi(tf); pi(tf))is only one among very many other microstates which all appear with the same probability.
As we will see, the number of microstates that is compatible with a given macroscopic observable is a quantityclosely related to the entropy of this macrostate. The larger , the more probable is the corresponding macrostate, and the macrostate with the largest number max of possible microscopic re-alizations corresponds to thermodynamic equilibrium. The irreversibility that comes wit the second law is essentially stating that the motion towards a state with large is more likely than towards a state with smaller .
Third law: This law was postulated by Nernst in 1906 and is closely related to quantum e¤ects at lowT. If one cools a system down it will eventually drop into the lowest quantum state. Then, there is no lack of structure and one expectsS !0. This however implies that one can not change the heat content anymore if one approaches T ! 0, i.e. it will be increasingly harder to cool down a system the lower the temperature gets.
2.2 Thermodynamic potentials
The internal energy of a system is written (following the …rst and second law) as
dU =T dS pdV + dN (2.20)
where we consider for the moment only one type of particles. Thus it is obviously a function
U(S; V; N) (2.21)
with internal variables entropy, volume and particle number. In particulardU = 0for …xedS,V, andN. In case one considers a physical situation where indeed these internal variables are …xed the internal energy is minimal in equilibrium.
Here, the statement of an extremum (minimum) follows from the conditions
@U
@xi = 0 with xi=S; V, orN: (2.22)
with
dU =X
i
@U
@xidxi (2.23)
follows at the minimum dU = 0. Of course this is really only a minimum if the leading minors of the Hessian matrix @2U=(@xi@xj) are all positive2. In addition, the internal energy is really only a minimum if we consider a scenario with …xedS,V, and N.
Alternatively one could imagine a situation where a system is embedded in an external bath and is rather characterized by a constant temperature. Then,U is not the most convenientthermodynamic potential to characterize the system.
A simple trick however enables us to …nd another quantity which is much more convenient in such a situation. We introduce the so calledfree energy
F=U T S (2.24)
which obviously has a di¤erential
dF =dU T dS SdT (2.25)
which gives:
dF = SdT pdV + dN: (2.26)
Obviously
F =F(T; V; N) (2.27)
i.e. the free energy has the internal variablesT,V andN and is at a minimum (dF = 0) if the system has constant temperature, volume and particle number.
The transformation fromU toF is called a Legendre transformation.
Of course,F is not the only thermodynamic potential one can introduce this way, and the number of possible potentials is just determined by the number of internal variables. For example, in case of a constant pressure (as opposed to constant volume) one uses theenthalpy
H =U+pV (2.28)
with
dH =T dS+V dp+ dN: (2.29)
If both, pressure and temperature, are given in addition to the particle number one uses thefree enthalpy
G=U T S+pV (2.30)
with
dG= SdT+V dp+ dN (2.31)
2Remember: LetHbe ann nmatrix and, for each1 r n, letHrbe ther rmatrix formed from the …rstrrows andrcolumns ofH. The determinantsdet (Hr)with1 r n are called the leading minors ofH. A functionf(x)is a local minimum at a given extremal point (i.e. @f =@xi= 0) inn-dimensional space (x= (x1; x2; ; xn)) if the leading minors of the HessianH =@2f =(@xi@xj)are all positive. If they are negative it is a local maximum.
Otherwise it is a saddle point.
2.2. THERMODYNAMIC POTENTIALS 11 In all these cases we considered systems with …xed particle number. It is however often useful to be able to allow exchange with a particle bath, i.e. have a given chemical potential rather than a given particle number. The potential which is most frequently used in this context is the grand-canonical potential
=F N (2.32)
with
d =SdT pdV N d : (2.33)
= (T; V; )has now temperature, volume and chemical potential as internal variables, i.e. is at a minimum if those are the given variables of a physical system.
Subsystems:There is a more physical interpretation for the origin of the Legendre transformation. To this end we consider a system with internal en-ergy U and its environment with internal energy Uenv. Let the entire system, consisting of subsystem and environment be closed with …xed energy
Utot=U(S; V; N) +Uenv(Senv; Venv; Nenv) (2.34) Consider the situations where all volume and particle numbers are known and
…xed and we are only concerned with the entropies. The change in energy due to heat ‡uxes is
dUtot=T dS+TenvdSenv: (2.35) The total entropy for such a state must be …xed, i.e.
Stot=S+Senv=const (2.36) such that
dSenv= dS: (2.37)
AsdU = 0by assumption for suc a closed system in equilibrium, we have
0 = (T Tenv)dS; (2.38)
i.e. equilibrium implies thatT =Tenv. It next holds
d(U +Uenv) =dU+TenvdSenv = 0 (2.39) wich yields for …xed temperatureT =Tenv and with dSenv= dS
d(U T S) =dF = 0: (2.40)
From the perspective of the subsystem (without …xed energy) is therefore the free energy the more appropriate thermodynamic potential.
Maxwell relations: The statement of a total di¤erential is very power-ful and allows to establish connections between quantities that are seemingly unrelated. Consider again
dF = SdT pdV + dN (2.41)
Now we could analyze the change of the entropy
@S(T; V; N)
@V (2.42)
with volume at …xedT andN or the change of the pressure
@p(T; V; N)
@T (2.43)
with temperature at …xed V and N. Since S(T; V; N) = @F(T ;V;N)@T and p(T; V; N) = @F(T ;V;N@V ) follows
@S(T; V; N)
@V = @2F(T; V; N)
@V @T = @2F(T; V; N)
@T @V
= @p(T; V; N)
@T (2.44)
Thus, two very distinct measurements will have to yield the exact same behav-ior. Relations between such second derivatives of thermodynamic potentials are called Maxwell relations.
On the heat capacity: The heat capacity is the change in heat Q=T dS that results from a change in temperaturedT, i.e.
C(T; V; N) =T@S(T; V; N)
@T = T@2F(T; V; N)
@T2 (2.45)
It is interesting that we can alternatively obtain this result from the internal energy, if measured not with its internal variables, but instead U(T; V; N) = U(S(T; V; N); V; N):
C(T; V; N) = @U(T; V; N)
@T
= @U(S; V; N)
@S
@S
@T = T@S
@T: (2.46)
2.2.1 Example of a Legendre transformation
Consider a function
f(x) =x2 (2.47)
with
df= 2xdx (2.48)
We would like to perform a Legendre transformation fromxto the variablep such that
g=f px (2.49)
and would like to show that
dg=df pdx xdp= xdp: (2.50)
2.3. GIBBS DUHEM RELATION 13