Statistical mechanics is actually part of Thermodynamics. The logical structure of thermodynamics can be shown as below,

Definitions: States refers to thermodynamic states, it does not refer to quantum states.. Kinetics refers to dynamics of molecules and atoms, etc.

Thermodynamic States refers to distributions of energy within a system (1). This can be applied to semiconductor physics (2), distribution of energy in electrons (primarily). Also laser physics (3), for the distrobution of energy among photons.

For Q quanta in N oscillators, the number of arrangements is W, expressed as,

We define density of states g(E) to allow to calculate many states for a given volume.

This clearly gives the number of states in the interval from E to E+dE. We might usually work with g(E) rather than W. Sometimes we work with ln(g(E)) if g(E) becomes too large.

Stirling’s approximation states the following,

n!=n.ln(n-1)
for the above example, g 10^{10^23}

It is impossible to answer the question of how energy is distributed due to many types of different distributions. However, time averages can be used to help determine energy distributions.

The probability that a system with energy, E, is in a state E_i is P_i.

P_i=N_i/N

Where N is the total number of experiments and N_i is the number of experiments in state i.

For a small number of discrete states,

 P_i = 1

For large W, we think of distributions of states.

dp=f(r).dr

Where f(r).dr is the probability of distribution functions, also,

dp = ,0(f(r).dr) = 1

These represent the probabilities of the distrobution of ditrobutions of energy.

The equilibrium state results in some observable property. e.g., Pressure, temperature, which do not change with time. We now ask what causes a state to change with time. We need to ask if the system is isolated or connected to another system. Note however, heat content is constant in a given state. This is an approximation however, there are actually small variations. We time average quantities to neglect this. Heat flows in or out of the system, the direction being determined by the second law of thermodynamics. We can precisely define what temperature is by using statistical mechanics, which shall be done later on in this course.

This is the flow of energy in thermodynamic systems. We know that,

Where,

• dE is the total change in internal energy.
• dq is the heat flowing in or out of the system.
• dw is the work done.

Internal energy is defined as the sum of particle energy and field energy. A reduction in feild energy leads to an increase in kinetic energy for all known forces in physics. This leads to an important conclusion. 1) The action of a force on particles does not change the internal energy. 2) To change the internal energy, heat must flow in or out or we must do work on a system.

External work alters the thermodynamic states (apperances might not be, in a homgeneous transparent liquid for example). Work done may be reversable or dissipative. In a dissipative process work done is converted into small scale motion processes. Energy is still conserved but is not recoverable. Such that we could say a change of volume does work dw=-P.dV. A change of constraint causes work to be done. A reversable process allows the energy to be lost and then returned or recovered, e.g. a system connected to a resivour. Fast processes are not usually reversable.

Transitions between systems

Consider a thermally isolated system. We don't know the quantum state of a large system, (thermal fluctuations - Heisenburg Uncertainty Principal).

We want to know the transition rate of i j. Assume P_i^~, P_j^~ are uniform over E. We also assume that E<E. The transition rate is governed by Fermi's golden rule. The One to Many jump rate is given by,

_BA=2|H_BA|²g_B/

H_BA is the matrix element for the transition.

If P_i^~, P_j^~ are uniform over E we can assume EE. The One to One jump rate is,

_ij=_BA/g_B.E

g_B.E gives you the number of states in the interval E.

E is known as the accessibility range.

Since we stated that EE, then E s also the accessibility range.

It can be shown that,

H_AB=H^*_BA

Therefore,
|H_BA|²=|H_AB|²

Therefore,
_ij=_ji

... in a thermally isolated system. This is the principal of detailed balance. eg, semiconductors in equilibrium, recombination=generation, i.e. balanced.

dp_i^~/dt = ij(p_j^~-p_i^~)
dp_i^~/dt = 0 in thermal equilibrium.

All states within the accessibility range E have equal probobility.

We now arrive at the meaning of temperature for large systems.

Consider two systems in thermal contact (A & B). They are free to exchange heat but otherwise weakly interacting. i.e. the quantum states are unaltered. We define a joint density of states across A&B.

The energy levels usually end up in a continuum. There is usually a bunching of states between constant energy surfaces.

We find that the joint density of states, g_A.g_B is maximised in the central region. Therefore, d(g_A.g_B)>0

This also means that,

d[ln(g_A.g_B)] = d[ln(g_A)] + d[ln(g_B)] > 0

Therefore,

(/E_A)(ln(g_A).dE_A + (/E_B)(ln(g_B)).dE_B > 0

dE_A = -dE_B (for constant energy)

[(/E_A)(ln(g_A) - (/E_B)(ln(g_B))].dE_A > 0

This inequality is crucial. If...

(/E_A)(ln(g_A)) < (/E_B)(ln(g_B))

Then,
dE_A < 0
Energy flows out of A into B.

Therefore (d/dE)(ln(g)) is a measure of coldness.

Therefore, we would say,

1/kT = (d/dE)(ln(g))
if k=1.38*10^-23 and T is absolute temperature.

It so happens, on the ideal gas scale, our k turns out to be identical to the Boltzman constant.

(see sheet "StatMech-01")

If we were to have a large system and a small system which are in thermal contact (weakly interacting otherwise) ("StatMech-01/B"). The small system has a large separation between states. There is no maximum in the joint state g_ag_b . The system always moves in the direction of increasing g_R.

From this we can say,

P_i g_R

In thermal equilibrium, temperature is constant.

Therefore,

1/kT = (ln(g_R)/E_R=contant
g_R exp(-E_R/kT)

Therefore,

P_i exp(-E_R/kT)

We know that P_i=1. Therefore,

P_i = (1/Z).exp(-E_i/kT)
Where Z=exp(-E_i/kT)

Z is a partition function

We should recognise this as as Boltzman distribution. The Boltzmann distribution defines the temperature for a small system (and in fact most small systems in thermal equilibrium with large systems).

But what about isolated systems? the density function g, depends on energy. Therefore the spacing between states becomes large.

In a small system we find that d(ln(g))/dE is discontinuous (i.e. it is not large enough to form a continuum). Thermal fluctuations can be so large that even energy is not well defined. Temperature of a small isolated system is not defined. However, for a small system in contact with a large system T is well defined. P_i becomes equal for states of low energy, in accordance with definition of thermal equilibrium. Therefore systems described by Boltzmann distrobution thend to be in their ground state.

This is defined as,

Where M is teh quantum multiplicity (ie the number of states over which energy is is spread).

If P_i is unfirom, then P=1/M.

Then S=-k.ln(P). Therefore if P=1, S=0, for joint systems, S=k.ln(M_A.M_B), S=S_A+S_B.

Idf Pi is not uniform,

S=-k.P_i.ln(P_i)

This effectively sib-divides the system into smaller systems where P is uniform of P_i.

S=-k.P_i.ln(P_i)

dS/dt = -k ln(P_i).(dP_i/dt)

Now for an isolated going from to

dP/dt = (P-P)

dP/dt = (P-P)

(P,<1)

dS/dt = k (ln(P)-ln(P))(P-P)

Therefore: dS/dt > 0

It is only in reversible processes that one can distinguish between heat and work.

In theses we have a slow change of constraint so no heating is caused. Work causes a change to constraints. i.e., work is being done. Intuitively, work corresponds to internal energy; heat corresponds to changes in occupancy of levels (states).

Suppose we have a system with 4 levels. We do work to move the levels but the occupancy does not change.

dW_rev=P_iE_i

Hence, dE = P_idE_i + E_idP_i

The second term is corresponding to reversible heat. This shows that for reversible heat flows into or out of a system we need the probabilities to change along side the energy states.

dE = dW_rev + dq_rev
dq_rev = E_idP_i

We want to show that dq_rev = T.dS. We know that ...

S=-kP_i.ln(P_i)

dS = -k [P_i d{ln(P_i)} + ln(P_i) dP_i]

We can further say that,

d{ln(P_i)}=(d{ln(P_i)}/dP_i).dP_i
d{ln(P_i)}=(1/P_i).dP_i

Hence we can say,

dS = -k [dP_i + ln(P_i) dP_i]

We know that P_i = 1, therefore it follows that dP_i=0.

Therefore,

dS = -k ln(P_i) dP_i

Assuming Boltzmann P_i=(1/Z).exp(-E_i/kT) we can also write,

ln(P_i) = -ln(Z)-(E_i/kT)

dS = -k (-(E_i/kT)-ln(Z))dP_i

dS = -k-dP_i(E_i/kT) + k.ln(Z)dP_i

Therefore,

dS = -k -dP_i(E_i/kT)

dS = dP_i(E_i/T)

We also know that TdS = Edp_i. Therefore,

TdS = dq_rev = EdP_i

This shows that dq_rev is the change in occupancy.

Consider a system with three levels and one quantum. Each level is separated by 0.05 ev. Work out the entropy at; 1)T=0, 2)T=400K, 3)T=1000K, 4)T=5000K, 5)T=infinity.

i.e. transfer of particles as well as energy. eg, semiconductor junctions. We know from these that the fermi levels equilibrate.

Consider an ideal system. This consists of a semi permeable membrane. It lets only one species of particle through. We will call this species S. There is also an ideal reservoir. An ideal reservoir consists of a large number of particles N all with energy E. So their total energy is NE. We will assume that the entropy does not change with particle number. This ideal system is isolated.

(S is entropy, Ns number of particles in system)
dS_TOT = (S/E)dE + (S/N_s)dN_s > 0

dE = -.dN, dS/dE=1/T

( is total energy per particle)

Therefore,

(dN_s/dt) + (S/N_s)dN_s 0

Define the chemiclal potential as:

µ_s/T = [S.N_s]_E,V

This is the total reversable work done per particle. Therefore,

(-µ_s)dN_s0

=> µ_s>

dN_s<0

Particle Exchange between two systems.

Here, we replace the reservoir by a real second system. We must consider change in entropy of the second system now.

dS_TOT = dS_A + dS_B

dS_TOT = (S/E_A)dE_A + (S/E_B)dE_B + (S/N_As)dN_As + (S/E_Bs)dE_Bs

Energy and particles are conserved!
Therefore,

dS_TOT=((1/T_A)-(1/T_B))d_EA + _s((-µ_s/T_A)+(µ_s/T_B))dN_As 0

(dE_A = -dE_B)


In equilibrium, S_TOT is maximised to changes in energy and particle number.

S_TOT/E_A = 0

S_TOT/E_B = 0

Which means that,

T_A=T_B

µ_A=µ_B

i.e. the chemical potentials are equalised in thermal equilibrium.

eg, Semiconductor-Metal junctions
Anderson Rule, - vacuum levels align.

Field Effect device (MOS capacitor)
In an ideal MOS capacitor

now with potential across it.

Let us now extend the previous treatment,

dS = (S/E)dE + (S/N_s)dN + [ (S/V)dV + similar constraint terms]

dS = dE/dT + µ_sdN_s/T + [P(dV/T) + ... ]

dE = TdS - [PdV + similar work terms] + µ_sdNs

We will use this in the derivation of Fermi-Dirac and Bose-Einstein
occupation functions

Indistinguishable Particles

In quantum mechanics we have a wave function , which is a mathematical expression of wave/particle duality. However, ||² represents the probability of finding the particle.
Suppose a pair of particles a,b.

(a,b)

If we swap theses particles around, we have the wave function

(b,a)

If a&b are indistinguishable, then,

|(a,b)|²=|(b,a)|²
then |(a,b)|=|(b,a)| : Symmetrical (bosons)
or |(a,b)|=-|(b,a)| : Antisymmetrical (fermions)

Suppose an assembly of similar but not identical particles. Individual particle states are (n₁=2, n₂=1, n₃=0, n₄=4 say)

(a,b,c,d) = ₁(a)₂(b)₂(c)₄(d)

For N distinct particle states, there are,

N!/(n₁!n₂!n₃!n₄!)

permutations.

Symmetry restricts the choices in reality. eg, exchange b&c.

'(a,b,c,d) = ₁(a)₁(c)₂(b)₄(d)

this wave function is not in reality different from . must be rewritten to allow for this. The rewritten version is known as a SYMMETRISED STATE.
eg,

=(1/2)[1(a)2(b)+1(b)2(a)]

a symmetrised state for two particles.

For symmetric particles there is only one state.
For antisymmetric |'|=-|| is not possible with symmetrised state. Therefore there is only one particle per state (which leads to things like Pauli exclusion principal).

Bose Einstein (BE) and Fermi Dirac (FD) Occupation functions

N_k fermions over g_k states.

_FD = _k (g_k!/N_k!(g_k-N_k)!)
_BE = _k ((N_k+g_k)/(N_k!g_k!))

Now use the TdS relationship, dE=TdS+µ_sdN_s
For S=1, then dE=TdS+dN (S=speicies)

dE/dN-µ=T(dS/dN)
S=k.ln()
(E-µ)/kT=(dln()/dN)

For fermions,

dln()/dN = ln((g_k-N_k)/N_k)
(from Stirling's approximation)

for bosons,

dln()/dN = ln ((g_k+N_k)/N_k)

ƒ_FD = 1/[exp((E-µ)/kT)+1]
ƒ_BE = 1/[exp((E-µ)/kT)-1]