LECTURE 2
afterthoughts from previous lecture
(i) the word "adiabatic" is ambiguous. Use "quasi-static" instead
(ii) the argument would work even for a single particle, for suitably shaped
cylinder for the piston to slide in.
(iii) A reference for that lecture, also (to some extent) the present one
is my review article Foundations of Statistical Mechanics in
Rep Prog Phys 42 (1979) 1937-2006
I Reversible and irreversible processes; entropy, temperature.
If by changing the control parameter x (e.g. the position of a piston)
you can get from state x = x1 to state x = x2 quasi-statically,
i.e. equilibrium is preserved as you go along, then there is a function
phi such that the process converts a microcanonical ensemble at
energy E1 with x = x1 to a m.c.e. at energy E2 = phi(E1) with x = x2.
If the process is reversible, then there is also a function taking
us back froom E2 to E1, and this function must be phi^{-1}
i.e. phi must be invertible
The argument I used before shows that W is an invariant for reversible
processes, i.e. W(E1,x1) = W(E2,x2) where x1 and x2 are the values of the
control parameter in the two states. This invariance can be used to
determine the function phi. See the example in Appendix 1 (not done in lecture)
For irreversible processes, E2 in general depends on the initial dynamical
state, but in some average sense it will be greater than phi(E1) and so
we expect W to increase in an irrev. process.
Entropy S is defined (here) by Boltzmann's formula (on his tombstone in Vienna)
S = k log W + a function of N which is not needed yet
where k:= Boltzmann's constant = 1.4 10^-23 J / K
So is an invariant for reversible processes (in a thermally isolated system)
and is expected to increase for an irreversible process.
The reason for the logarithm is that it makes the entropy additive:
if you put have a system consisting of two separate parts its W function is
W(E1,E2) = W1(E1) W2(E2), so the entropies add.
II Weakly interacting subsystems
Suppose a system consists of two almost non-interacting subsystems
H(omega) = H1(omega1) + H2(omega2) + weak interaction
The weak interaction is supposed to be large enough to destroy
the constancy of H1 and H2, but small enough for its numerical
value to be neglected in comparison with H1 and with H2
For example the two subsystems could be a hot and a cold body
brought into contact, or H1 could be a single molecule in a gas
and H2 the remaining molecules. I will work out the probability
distribution of subsystem 1 in its own phase subspace,
assuming that the compositie system is microcanonically distributed.
The probability "density" in the composite phase space is
rho(omega) = delta(E - H(omega1) - H2(omega2))/W'(E)
The probability density for subsystem 1 can be found by
calculating the expectation of an arbitrary dynamical variable G(omega1)
relating to that subsystem G(omega1), which is
int d omega1 int d omega2 G(w1) delta(E - H1(omega1) - H2(omega2))/W'(E)
= int d omega1 g(omega1) W2'(E - H1(omega1))/W'(E)
on carrying out the integral with respect to omega2 and using the definition
of W2'. Thus the probability density in the phase space of subsystem 1 is
(1) W2'(E - H1(omega1))/W'(E)
III Structure function for an ideal monatomic gas; Maxwell's distribution
This expression can be simplified if subsystem 1 is only a small part of the
whole system, e.g. when the entire system is an ideal monatomic gas
and H1 is one molecule of that gas. For an ideal monatomic gas the
Hamiltonian is
H(omega) = sum_i p_i^2/2m if all particles are in the box
= + infty if any particle is not in the box \Lambda
The structure function is therefore
W(E) = int_Omega d^{3N}p_i d^{3N}q_i step (E - H(omega))
= int_{Lambda^N} d^{3N}q_i int_{R^{3N}} step(E - sum_i p_i^2/2m)
= V^N times volume of a sphere of radius sqrt(2mE) in 3N/2 dimensions
= V^N (2 pi m E)^{3N/2} / (3N/2)!
where (3N/2)! := Gamma(3N/2 + 1). Using this formula to evaluate W2', we find
that the dependence of the prob. density (1) on H1(\omega1) is
W2(E - H1(omega1)) \propto (E - H1(omega1))^{3(N-1)/2 - 1}
\propto exp ( - beta p_1^2/2m + ...)
where
beta = (3(N-1)/2 - 1)/E \approx 3N/2E if N is very large
and I took the number of particles in subsystem 2 to be N-1.
This Gaussian distribution of single-particle momenta is called
the Maxwellian distribution (reference to be supplied, e.g.
his Collected Works).
IV The Gibbs distribution, heat baths, temperature
More generally, system 2 could be any large system, then (1)
can be simplified, using Taylor's expansion in the exponent
W2'(E-H1) = exp [log W2'(E-H1)]
= exp [log W2'(E) - H1 W2"(E)/W2'(E) + ...]
\propto exp [ - beta_1 H1(omega)]
where beta_1 := W2"(E)/W2'(E). This distribution applies to any system
immersed in a heat bath, i.e. a system much larger than itself
whose temperature is not affected by changes in energy of the
smaller susbsystem. It is called the Gibbs distribution or
canonical ensemble.
[the following was not covered properly in the lecture:
To relate beta_1 to the entropy, define
beta := d log W /dE = (1/k) dS/dE by Boltzmann's formula
so that W'(E) = beta W(E), log W'(E) = log beta + logW(E). A second
differentiation gives
beta_1 = d (log beta) / dE + beta
If E is large, the E in the denominator makes d(log beta) /dE << beta,
so that to a good approximation
beta_1 = beta
The temperature T is defined by
1/T = dS/dE, i.e. T = dE/dS
so that the definition of beta can be written
beta = 1/kT
and the Gibbs distribution can be written exp (- H(omega1)/kT).
V The thermodynamic limit, Ruelle's theorem
The normalization constant for the canonical distribution is
N! Z, where Z is the canonical partition function defined by
Z = (1/N!)int_\Omega exp ( - beta H(omega))
the factor (1/N!) is inserted by convention, so as to make the
asymptotic behaviour of Z in the thermodynamic limit simpler.
By the thermodynamic limit we mean a limit in which the system
becomes very large but the 'intensive' properties such as
denstity and energy density are held fixed or approach finite limits.
That is, in this limit
E,V,N (the so-called extensive variables) become large
but all their ratios such as E/V, N/V etc approach finite limits.
A method discovered by Ruelle (Helv. Physica Acta 36 (1963) 183)
makes it possible to show that certain thermodynamic functions such
as the entropy are also extensive, i.e. S becomes large but S/V
approaches a finite limit. Ruelle's method applies to the thermodynamic
function the free energy, which is related to the partition function by
F(T, V, N) = - kT log Z
and to the entropy by
F = E - TS
A very good account of Ruelle's method, with some extra results,
is M E Fisher 'The free energy of a macroscopic system'
Arch Rat Mech Anal 17 (1964) 377-410, but both these references do not
directly deal with the thermodynamic limit of the most basic
thermodynamic function, entropy. The entropy is done in
Ruelle's book Statistical Mechanics (Benjamin 1969).
A simple model for which Ruelle's theorem can be applied
is a system of hard spheres (non-rotating) which interact by
a two-body attractive potential falling off faster than
the inverse cube of the distance.
H(omega) = sum_i p_i^2/2m + sum_{i a,
epsilon being some positive number. The container is taken to be
a cube Lambda and the wall term is defined to be zero if all of the
spheres are inside Lambda but + infinity otherwise.
The the simplest version of the theorem says that for any sequence Lambda_k
of cubes each one of which has twice the side of its predecessor,
the following limit exists
- beta f(rho, beta) = lim sup_{k -> infty} V_k^{-1} log Z(\Lambda, N_k, beta)
where V_k = 8^k V_0 is the volume of the kth cube,
N_k := 8^k N_0 , rho := N_k/V_k = N_0/V_0.
The theorem also shows that f is a concave function of 1/beta and
convex function of rho, and that f(rho,beta)/rho is a convex decreasing function of 1/rho.
Another way of stating the result: if we define the free energy F by
F(V,N,T) := - kT log Z(V,N,1/kT)
then the theorem tells us that , in the thermodynamic limit
F(V,N,T) \sim V f(N/V,1/kT).
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Appendix 1: example of how to get the function phi relating
two states connected by a reversible process.
We know from section III above that in an ideal monatomic gas
W(E,V) = N log V + (3N/2) log E = N log (VE^{3/2})
If the states (E1,V1) and (E2,V2) are relatedby a reversible process,
we must have W(E1,V1) = W(E2,V2), which implies
V1 E1^{3/2} = V2 E2^{3/2} i.e. E2/E1 = (V2/V1)^{2/3}
Solving for E2 we get E2 = (V1/V2)^{2/3} E1 and the function
phi is phi: phi(t) = (V1/V2)^{2/3} t
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Appendix 2: outline of proof of the above theorem of Ruelle
It uses two ingredients deriving from the assumptions about the interaction:
(i) stability: U > - A - BN for some positive constants A, B
(ii) U <= U(particles in Lambda1) + U(particles in Lambda2)
From (i), Z < (e/N)^N (2 pi m/beta)^(3N/2) V^N exp beta (A + BN)
so (log Z)/V is bounded above
(ii) Z(Lambda1 + Lambda2, N1 + N2, beta) >
(Lambda1, N1, beta)Z(Lambda2, N2, beta)
so that log Z/V cannot decrease
Hence it approaches a limit as V -> infty.