Introduction

• Thermodynamics: phenomenological description of equilibrium bulk properties of matter in terms of only a few “state variables” and thermodynamical laws.

• Statistical physics: microscopic foundation of thermodynamics

• ∼ 10^{23} degrees of freedom → 2–3 state variables!

• “Everything should be made as simple as possible, but no simpler” (A. Einstein)

Summary of contents:

• Review of thermodynamics

• Thermodynamical potentials

• Phase space and probability

• Quantum mechanical ensembles

• Equilibrium ensembles

• Ideal fluids

• Bosonic systems

• Fermionic systems

• Interacting systems

• Phase transitions and critical phenomena

1. Foundations of thermodynamics

1.1. Fundamental thermodynamical concepts

System : macroscopic entity under consideration.

Environment : world outside of the system (infinite).

Open system : can exchange matter and heat with the environment.

Closed system : can exchange heat with the environment while keeping the number of particles fixed.

Isolated system : can exchange neither matter nor heat with the environment. Can (possibly) still do work by e.g. expanding.

Thermodynamical equilibrium: • No macroscopic changes.

• Uniquely described by (a few) external variables of state.

• System forgets its past: no memory effects, no hysteresis.

• Often the term global equilibrium is used, as opposed to local equilibrium, which is not full equilibrium at all (next page)!

Nonequilibrium:

• Generally much more complicated than equilibrium state.

• Simplest case: isolated systems each in an equilibrium state.

• In a local thermodynamical equilibrium small regions are locally in equilibrium, but neighbour regions in different equilibria ⇒ particles, heat etc. will flow. Example: fluid (water) with non-homogeneous temperature.

• Stronger nonequilibrium systems usually relax to a local equilibrium.

Degrees of freedom (d.o.f.) is the number of quantities needed for the exact description of the microscopic state.

Example: classical ideal gas with N particles: 3N coordinates (x, y, z), 3N momenta (p_{x}, p_{y}, p_{z}).

State variables are parameters characterizing the macroscopic thermodynamical state. These are all extensive or intensive:

Extensive variable: change value when the size (spatial volume and the number of degrees of freedom) is changed: volume V, particle number N, internal energy U, entropy S, total magnetic moment ∫ d^{3}r M.

Intensive variable: independent of the size of the system, and can be determined for every semimicroscopical volume element: e.g. temperature T, pressure p, chemical potential μ, magnetic field H, ratios of extensive variables like ρ = N/V, s = S/N, . . ..

Conjugated variables: A and B appear in pairs in expressions for the differential of the energy (or more generally, some state variable), i.e. in forms ±A dB or ±B dA; one is always extensive and the other intensive. Example: pressure p and volume V; change in internal energy U when V is changed (adiabatically, at constant S) is dU = −pdV.

Process is a change in the state.

Reversible process: advances via states infinitesimally close to equilibrium, quasistatically (“slow process”). The direction of a reversible process can be reversed, obtaining the initial state (for system + environment!)

Isothermal process : T constant.

Isobaric process : p constant.

Isochoric process : V constant.

Isentropic or adiabatic process: S constant.

Irreversible process is a sudden or spontaneous change during which the system is far from equilibrium. In the intermediate steps global state variables (p, T, …) are usually not well defined.

Cyclic process consists of cycles which take the system every time to its initial state.

1.2. State variables and exact differentials

Let us suppose that, for example, the state of the system can be uniquely described by state variables T, V ja N. Other state variables are then their unique functions:

p = p(T, V, N)
U = U(T, V, N)
S = S(T, V, N) …

By applying differential calculus, the differential of p, for example, is

dp = (\frac{∂p}{∂T} )_{V,N} dT + (\frac{∂p}{∂V} )_{T,N} dV + (\frac{∂p}{∂N} )_{T,V} dN
⋮

The differentials of state variables, dp, dT, dV, …, are exact differentials. These have the following properties

(A) Their total change evaluated over a closed path vanishes:

∮_{1→2} dp = ∮_{1→2} dU = ⋯ = 0.

(B) The total change of an exact differential is independent on the path of integration:

∫_{a} dU − ∫_{b} dU = 0,

so that we can write

U(2) = U(1) + ∫_{1}^{2} dU

Exact differentials

Let us denote by d̄F a differential which is not necessarily exact (i.e. integrals can depend on the path). Assuming it depends on 2 variables x, y, the differential

d̄F = F_{1}(x, y) dx + F_{2}(x, y) dy

is exact differential if

\frac{∂F_{1}}{∂y} = \frac{∂F_{2}}{∂x} .

Then ∃F (x, y) so that F_{1}(x, y) = \frac{∂F (x,y)}{∂x} and F_{2}(x, y) = \frac{∂F (x,y)}{∂y} and

∫_{1}^{2} d̄F = F (2) − F (1)

is independent on the path, and integrable. In this case (x, F_{1}) and (y, F_{2}) are pairs of conjugated variables with respect to F.

Examples: are the following differentials exact?

d̄F = y dx + x dy
d̄F = x dx + x dy

All physical state variables are exact differentials! This will enable us to derive various identities between state variables.

Integrating factor

If d̄F = F_{1}dx + F_{2}dy is not exact, there exists an integrating factor λ(x, y) so that in the neighbourhood of the point (x, y)

λd̄F = λF_{1}dx + λF_{2}dy = df

is an exact differential. λ and f are state variables.

Example: find λ for the differential

d̄F = x dx + x dy .

Legendre transformations

Legendre transformations can be used to make changes in the set of the indepependent state variables. For example, let us look at the function f(x, y) of two variables. We denote

z = f_{y} = \frac{∂f(x, y)}{∂y}

and define the function

g = f − yf_{y} = f − yz.

(Note: z, y is a congjugated pair with respect to f!) Now

dg = df − y dz − z dy = f_{x}dx + f_{y}dy − y dz − z dy
= f_{x}dx − y dz.

Thus we can take x and z as independent variables of the function g, i.e. g = g(x, z). Obviously

y = −\frac{∂g(x, z)}{∂z} .

Corresponding to the Legendre transformation f → g there is the inverse transformation g → f

f = g − zg_{z} = g + yz.

Often needed identities

Let F = F (x, y), x = x(y, z), y = y(x, z) and z = z(x, y). If we want to give F in terms of (x, z), we can write

F (x, y) = F (x, y(x, z)).

Applying differential rules we obtain identities

(\frac{∂F}{∂x} )_{z} = (\frac{∂F}{∂x} )_{y} + (\frac{∂F}{∂y} )_{x} (\frac{∂y}{∂x})_{z}

(\frac{∂F}{∂z} )_{x} = (\frac{∂F}{∂y} )_{x} (\frac{∂y}{∂z})_{x}

One can show that

(\frac{∂x}{∂y})_{z} = \frac{1}{(\frac{∂y}{∂x})z}

(\frac{∂x}{∂y})_{z} (\frac{∂y}{∂z})_{x} (\frac{∂z}{∂x})_{y} = −1

and

(\frac{∂x}{∂y})_{z} = \begin{matrix}(\frac{∂F}{∂y} ) & z \\ (\frac{∂F}{∂x} ) & z \end{matrix}.

1.3. Equations of state

Encodes (some of the) physical properties of the equilibrium system. Usually these relate “mechanical” readily observable variables, like p, T, N, V; not “internal” variables like S, internal energy U etc. A typical example: pressure of some gas as a function of T and density ρ.

Some examples:

Classical ideal gas

pV = Nk_{B}T

where N = number of molecules T = absolute temperature k_{B} = 1.3807 · 10^{−23} J/K = Boltzmann constant . Chemists use often the form

pV = nRT
n = N/N_{0} = number of moles
R = k_{B}N_{0} = 8.315J/K mol
= gas constant
N_{0} = 6.0221 · 10^{23} = Avogadro’s number .

If the gas is composed of m different species of molecules the equation of state is still

pV = Nk_{B}T,

where now

N = m ; ∑ ; i=1 N_{i}

and

p = _{i}^{∑} p_{i}, p_{i} = N_{i}k_{B}T/V,

where p_{i} is the partial pressure of the i:th component

Virial expansion of real gases

When the interactions between gas molecules are taken into account, the ideal gas law receives corrections which are suppressed by powers of density ρ = N/V:

p = k_{B}T [ρ + ρ^{2}B_{2}(T) + ρ^{3}B_{3}(T) + ⋯]

Here B_{n} is the n:th virial coefficient.

Van der Waals equation

The molecules of real gases interact

• repulsively at short distances; every particle needs at least the volume b ⇒ V ∼^{>}Nb.

• attractively (potential ∼ (r/r_{0})^{6}) at large distances due to the induced dipole momenta. The pressure decreases when two particles are separated by the attraction distance. The probability of this is ∝ (N/V)^{2}.

We improve the ideal gas state equation

p′V′ = Nk_{B}T

so that

V′ = V − Nb
p = p′ − aρ^{2} = true pressure .

then

(p + aρ^{2})(V − Nb) = Nk_{B}T.

Solid substances The thermal expansion coefficient

α_{p} = \frac{1}{V} (\frac{∂V}{∂T} )_{p,N}

and the isothermal compressibility

κ_{T} = −\frac{1}{V} (\frac{∂V}{∂p} )_{T,N}

of solid materials are very small, so the Taylor series

V = V_{0}(1 + α_{p}T − κ_{T} p)

is a good approximation. Typically

κ_{T} ≈ 10^{−10} /Pa
α_{p} ≈ 10^{−4}/K.

Stretched wire

Tension [N/m^{2}]

σ = E(T)(L − L_{0})/L_{0},

where L_{0} is the length of the wire when σ = 0 and E(T) is the temperature dependent elasticity coefficient.

Surface tension

σ = σ_{0} (1 − \frac{t}{t′})^{n}
t = temperature ^{∘}C
t′ and n  experimental constants ,

1∼^{<}n∼^{<}2

σ_{0} = surface tension when t = 0^{∘}C.

Electric polarization When a piece of material is in an external electric field E, we define

D = ϵ_{0}E + P,

where

P = electric polarization
= atomic total dipole momentum/volume
D = electric flux density
ϵ_{0} = 8.8542 · 10^{−12} As/Vm
= vacuum permeability .

In homogenous dielectric material one has

P = (a + \frac{b}{T} ) E,

where a and b are almost constant and a, b ≥ 0.

Curie’s law

When a piece of paramagnetic material is in magnetic field H we write

B = μ_{0}(H + M),

where

M = magnetic polarization
= atomic total magnetic moment/volume
B = magnetic flux density
μ_{0} = 4π · 10^{−7}Vs/Am = vacuum permeability .

Polarization obeys roughly Curie’s law

M = \frac{ρC}{T}H,

where ρ is the number density of paramagnetic atoms and C an experimental constant related to the individual atom.

Note Use as a thermometer: measure the quantity M/H.

1.4. 0th law of thermodynamics

If each of two bodies is separately in thermal equilibrium with a third body then they are also in thermal equilibrium with each other ⇒ there exists a property called temperature and thermometer which can be used to measure it.

1.5. Work

Work is exchange of such ”noble” energy (as opposed to exchange of heat or matter) that can be completely transformed to some other noble form of energy; e.g. mechanical and electromagnetic energy.

Sign convention: work ΔW is the work done by the system to its environment.

Example pV T system

ΔW = p ΔV.

Note d̄W is not an exact differential: the work done by the system is not a function of the final state of the system (need to know the history!). Instead

\frac{1}{p}d̄W = dV

is exact, i.e, 1/p is the integrating factor for work.

Example

d̄W = p dV − σA dL − E · dP − H · dM.

In general

d̄W = _{i}^{∑} f_{i}dX_{i} = f · dX,

where f_{i} is a component of a generalized force and X_{i} a component of a generalized displacement.

1.6. 1st law of thermodynamics

Total energy is conserved

In addition to work a system can exchange heat (thermal energy) or chemical energy, associated with the exchange of matter, with its environment. Thermal energy is related to the energy of the thermal stochastic motion of microscopic particles.

The total energy of a system is called internal energy.

Sign conventions:

[figure]

ΔQ

ΔW

[figure]

μ ΔN

chemical energy System

Environment

If the system can exchange heat and particles and do work, the energy conservation law gives the change of the internal energy

dU = d̄Q −d̄W + μdN,

where μ is the chemical potential. More generally,

dU = d̄Q − f · dX + _{i}^{∑} μ_{i}dN_{i}.

U is a state variable, i.e. dU is exact.

Cyclic process

In a cyclic process the system returns to the original state. Now ∮ dU = 0, so ΔW = ΔQ (no change in thermal energy). In a pV T -system

[figure: p; V; ΔQ^{+}; ΔW = ∮ p dV = Area!; ΔQ^{−}]

The total change of heat is

ΔQ = ΔQ^{+} + ΔQ^{−},

where ΔQ^{+} is the heat taken by the system and −ΔQ^{−} (> 0) the heat released by the system.

The efficiency η is

η = \frac{ΔW}{ΔQ^{+}} = \frac{ΔQ^{+} + ΔQ^{−}}{ΔQ^{+}} = 1 − \frac{|ΔQ^{−}|}{|ΔQ^{+} |} .

1.7. 2nd law of thermodynamics

Heat flows from high temperatures to low temperatures.

(a) Heat cannot be transferred from a cooler heat reservoir to a warmer one without other changes.

(b) In a cyclic process it is not possible to convert all heat taken from the hotter heat reservoir into work.

(c) It is not possible to reverse the evolution of a system towards thermodynamical equilibrium without converting work to heat.

(d) The change of the total entropy of the system and its environment is positive and can be zero only in reversible processes.

(e) Of all the engines working between the temperatures T_{1} and T_{2} the Carnot engine has the highest efficiency.

We consider the infinitesimal process

[figure: y; x; 1; U(1); 2; U(2); d̄Q]

Now

d̄Q = dU +d̄W = dU + f · dX,

so there exists an integrating factor 1/T so that

\frac{1}{T} d̄Q = dS

is exact. The state variable S is entropy and T turns out to be temperature (on an absolute scale)

The second law (d) can now be written as

\frac{dStot}{dt} ≥ 0,

where S_{tot} is the entropy of the system + environment.

For the entropy of the system only we have

dS ≥ \frac{1}{T} d̄Q,

where the equality holds only for reversible processes. The entropy of the system can decrease, but the total entropy always increases (or stays constant).

For reversible processes the first law can be rewritten as

dU = d̄Q −d̄W + μ dN = T dS − p dV + μ dN.

1.8. Carnot cycle

Illustrates the concept of entropy. The Carnot engine C consists of reversible processes

a) isothermal  T_{2}  ΔQ_{2} > 0
b) adiabatic  T_{2} → T_{1} ΔQ = 0
c) isothermal T_{1} ΔQ_{1} > 0
d) adiabatic  T_{1} → T_{2} ΔQ = 0

Now ΔU = 0, so ΔW = ΔQ_{2} − ΔQ_{1} (wrong sign here, for simplicity).

We define the efficiency as

η = \frac{ΔW}{ΔQ_{2}} = 1 − \frac{ΔQ_{1}}{ΔQ_{2}} .

Because the processes are reversible the cycle C can be reversed and C works as a heat pump.

Let us consider two Carnot cycles A and B, for which

ΔWA = ΔWB = ΔW.

A is an enegine and B a heat pump. The efficiences are correspondingly

ηA = \frac{ΔW}{ΔQA} and ηB = \frac{ΔW}{ΔQB} .

Let us suppose that

ηA > ηB,

so that ΔQB > ΔQA or ΔQB − ΔQA > 0. The heat would transfer from the cooler reservoir to the warmer one without any other changes, which is in contradiction with the second law (form a). So we must have

ηA ≤ ηB.

By running the engines backwards one can show that

ηB ≤ ηA,

so that ηA = ηB, i.e. all Carnot engines have the same efficiency.

Note The efficiency does not depend on the realization of the cycle (e.g. the working substance) ⇒ The efficiency depends only on the temperatures of the heat reservoirs.

Similarly, one can show that the Carnot engine has the highest efficiency among all engines (also irreversible) working between given temperatures.

Let us consider Carnot’s cycle between temperatures T_{3} and T_{1}. Now

η = 1 − f(T_{3}, T_{1}),

where

f(T_{3}, T_{1}) = \frac{ΔQ_{1}}{ΔQ_{3}} .

Here

f(T_{3}, T_{2}) = \frac{ΔQ_{2}}{ΔQ_{3}}
f(T_{2}, T_{1}) = \frac{ΔQ_{1}}{ΔQ_{2}}
f(T_{3}, T_{1}) = \frac{ΔQ_{1}}{ΔQ_{3}}

so

f(T_{3}, T_{1}) = f(T_{3}, T_{2})f(T_{2}, T_{1}).

The simplest solution is

f(T_{2}, T_{1}) = \frac{T_{1}}{T_{2}}.

We define the absolute temperature so that

η = 1 − \frac{T_{1}}{T_{2}}.

The Carnot cycle satisfies

∮ \frac{d̄Q}{T} = 0,

since

∫_{a} \frac{d̄Q}{T} = \frac{ΔQ_{2}}{T_{2}}

and

∫_{c} \frac{d̄Q}{T} = −\frac{ΔQ_{1}}{T_{1}} = −\frac{ΔQ_{2}}{T_{2}} .

This is valid also for an arbitrary reversible cycle because

∮_{C} \frac{d̄Q}{T} = _{i}^{∑} ∮_{C_{i}} \frac{d̄Q}{T} = 0.

So

dS = \frac{d̄Q}{T}

is exact and the entropy S is a state variable.

Because the Carnot cycle has the highest efficiency a cycle containing irreversible processes satisfies

η_{irr} = 1 − \frac{ΔQ_{1}}{ΔQ_{2}} < η_{Carnot} = 1 − \frac{T_{1}}{T_{2}}

or

\frac{ΔQ_{2}}{T_{2}} − \frac{ΔQ_{1}}{T_{1}} < 0.

Thus for an arbitrary cycle we have

∮ \frac{d̄Q}{T} ≤ 0,  (∗)

where the equality holds only for reversible processes.

For an arbitrary process 1 → 2 the change of the entropy can be obtained from the formula

ΔS = ∫_{rev} dS = ∫_{rev} \frac{d̄Q}{T}.

According to the formula (∗) we have

∫_{irr} \frac{d̄Q}{T} − ∫_{rev} \frac{d̄Q}{T} < 0,

or

ΔS > ∫_{irr} \frac{d̄Q}{T}.

This is usually written as

dS ≥ \frac{d̄Q}{T}

and the equality is valid only for reversible processes.

In an isolated system we have

ΔS ≥ 0.

1.9. 3rd law of thermodynamics

Nernst’s law:

lim_{T →0} S = 0.

A less strong form can be stated as:

When the maximum heat occuring in the process from a state a to a state b approaches zero the also the entropy change ΔS_{a→b} → 0.

Note There are systems whose entropy at low temperatures is larger than true equilibria would allow.