Statistical Mechanics

Thermodynamics

Thermodynamics, essentially, is a phenomenological description of macroscopic system in thermal equilibrium.

How to idealize the system

View it as some ensemble such as:

• closed system: often descripted as a closed box with adiabatic walls. (Microcanonical)
• open system: with exchanges
  • canonical: exchanges heat. (Two system with diabatic wall)
  • grandcanonical: exchanges both heat and particles, etc.

How to describe the system

• State function
• Thermodynamics coordinates.
• State Variable: (Pressure), (Volume), (Chemical Potential), etc.

The idea of SM Thermodynamics(example):

For kinetic theory of gas, the system in micro perspective can be completely described as as a point in phase space of dim (3 for 3d).

And in phase space, a probability measure is introduced for each state.

The Boltzmann equation and H theorem says that the equilibrium we can observe occurs as the maximization of entropy

Use Lagrangrian Multipler:

Define to absorb .

Question to be answered:

• Why the equilibrium occurs when is maximized (and why is in this form?)
• How to calculate the amount we care from this? (To be precise, ensemble averages )

We will go back to this later, but now we shall first look at the classical thermodynamics.

The Zeroth Law of Thermodynamics:

The system is usually of particles and the possible states are at least of . How we can use a marco parameter such as temperature to describe a great portion of such possible states?

In common sense, marco parameters are a property shared by physical systems. To make this statement well-defined, we should introduce the zeroth law of Thermodynamics:

(The transitivity of equilibrium) if system A is in equilibrium with system B and system B is in equilibrium with system C. Then A is in equilibrium with system C.

Therefore, a system in thermoequilibrium can be described in a state function that is purely described by its thermocoordinate (things that equals when in thermoequilibrium), as equivalence class in math.

The First Law of Thermodynamics

The amount of work required to change the state of an otherwise adiabatically isolated system depends only on the initial and final states, and not on the means by which the work is performed, or on the intermediate stages through which the system passes.

For the thermodynamics system we can construct a state function, the internal energy . The amount of can be derived from amount of work needed for an adiabatic transformation from an inital state to a final state using .

What if the transformation is not adiabetic (which is the most of the cases)?

It is observed that , We call the extra part that cannot be explained by as

Sometimes we use generalized force and displacement to charactize the work done:

System	Force	Displacement
Wire	tension	length
Fluid	pressure	volume
Chemical reaction	chemical potential	particle number

Ideal gas: T does not change during expansion.

Response functions:

Charactizing the macroscopic behaviour of the system:

Heat capacities:

For ideal gas

Isothermal compressibility:

The Second Law of Thermodynamics

The impetus for the second law of thermodynamics is the advent of heat engines.

Heat engine: a device that receives energy in the form of heat from a hot reservoir, delivers work to its surroundings, and discharges energy in the form of heat to a cold reservoir.

Heat pump: a device that uses work to delivers energy in the form of heat to a hot reservoir, and receives energy in the form of heat from a cold reservoir.

-------                      ------
|Source|--Qh->|Engine|--Q_c->|Sink|
-------         |            ------
                |
                v
               Work

               Work
                |
                |
                v
-------                       -------
|Icebox|--Qc->|Fridge| --Qh->|Exhaust|
-------                       ------- 

Heat engine efficiency:

Heat pump efficiency:

The second law of thermodynamics states that the efficiency of a heat engine can never be 100%.

Kelvin’s Statement: No process is possible whose sole result is the complete conversion of heat into work.

Clausius’s Statement: No process is possible whose sole result is the transfer of heat from a cooler to a hotter body.

The best heat engine is the Carnot engine, which consists of two isothermal processes and two adiabatic processes.

1. Isothermal expansion: the system is in contact with the hot reservoir at temperature , and expands from volume to .
2. Adiabatic expansion: the system is isolated and expands from volume to .
3. Isothermal compression: the system is in contact with the cold reservoir at temperature , and compresses from volume to .
4. Adiabatic compression: the system is isolated and compresses from volume to .

See https://galileo.phys.virginia.edu/classes/152.mf1i.spring02/CarnotEngine.htm for more details.

We argue that the efficiency of the Carnot engine is the upper bound of all heat engines, otherwise we can use the more efficient engine to run the Carnot engine in reverse to create a perpetual motion machine of the second kind (Pump heat from cold to hot without work input).

Use the ideal gas as an example, we can calculate the efficiency of the Carnot engine is Therefore, the efficiency of any heat engine is bounded by the efficiency of the Carnot engine:

The Third Law of Thermodynamics

For an infinismal process that draw from a heat reservoir at temperature

W.r.t. the whole system:

We can decompose any process into a series of infinitismal processes, and for each process, the system exchange heats with a system with almost identical temperature. We know that: where is the temperature of the system during the -th process. Therefore, we have:

Which leads to Clausius’s theorem

For a reversible cycle, we infer that . And we can define a function of state , only dependent on two end-points:

Kinetic Theory of Gas

Liouville Theorem

Since . So .

The time evolution of the ensemble average

Now, consider the equilibrium state: A possible solution is constant on constant energy surface, etc. or for additional conserved quantities.

non-stationary densities converge onto the stationary solution

Actually, this is the fact that the solutions are in the neighborhood of for the most of the time.

Need ergodicity to justify.

BBGKY Hierarchy

Define s-partial density Assume the Hamiltonian

BBGKY Formula

or Note that the LHS is the full time derivative of , or action under adjusted by a scatter term (RHS).

---

Boltzmann Equation & H-Theorem

First Level in BBGKY equation

Estimation

LHS of RHS of LHS of RHS of

So we can set RHS of to 0.

And assume when distant So since the position won’t change so much.

Therefore, substitute into RHS of :

Change variable with , , .

Since term and term is small, now be the coord along the line.

coll_illustration

We conclude that

H Theorem

By the Livouille Theorem, we can swap and , and . By the symmetricity of , we can swap the subscript too.

Equilibrium properties

A necessary condition for is that at each point We observed additive conserved quantities in the collision.

1. Number of particles
2. Momentum
3. Energy Or with potential energy Assume uniform, , only depend on or any other quantity that is conserved by it . For example, as long as , independent of and .

Assume .

Equilibrium between two gases:

Equilibrium, all right assume to be zero.

Can be satisfied with Ideal Gas Equation

Conservation Law

Stage 1:

Governed by the fast collision term in the right of Boltzmann equation. (reached zero) equilibrium.

local density, governed by the local conservatives.

Stage 2:

Governed by the streaming term

It is most conveniently expressed in terms of the time evolution of conserved quantities according to hydrodynamic equations.

Conserved quantity

If conserved in collision does not change.

Now for the left term: Particle Number ()

Momentum ()

If : Kinetic Energy

Plug in the peculiar speed

Zeroth-Order Solution

Assume is in local equilibrium:

This satisfies the collision term exactly (RHS of Boltzmann = 0), but does not satisfy the full Boltzmann equation because the streaming operator acts on the slowly varying fields .

First order equation

(Classical) Statistical Mechanics

The idea is that we start from the simplest situation (microcanonical). Then we gradually allow extensive quantity to fluctuate but add intensive quantity in control.

Microcanonical System

Particles system where is fixed.

The system is equally probable in any state that satisfy the energy constraint.

Canonical System

We now fix instead of for a system in contact with a heat reservoir.

Consider the whole system , where is the heat reservior, is the canonical system we want to study.

For the whole system , it is in the microcanonical ensemble.

The last step is expand at to the first order, and use .

Question:

Why not expand directly? Hint: is exponential in . Draw ’s expansion at to see the difference.

Grandcanonical System

We now fix instead of .

For similar reason, the system can be in energy , particle number with probability:

Gibbs System

Fix , change to , to .

Partition Function and the Calculations

Here we assume the hamiltonian is non-interacting particles in a box of volume and potential energy .

where is unnormalized distribution.

In microcanonical ensemble:

In canonical ensemble:

The Legrendre transform from to (T) correspond to the Laplace transform in the partition function.

The integration formula for Gamma function:

The average energy is given by: Or

we can also calculate the one particle partition function : where is thermal de Broglie wavelength.

In grandcanonical ensemble:

In Gibbs ensemble:

Ensembles

Reversible Process

Sometimes we use be the Shannon entropy of the system, be the average energy.

The advantage is that is dimensionless, and is in energy unit .

Definition:

A process is reversible if carried out in such a way that the system is always infinitesimally close to the equilibrium condition.

If it is not in a equilibrium state, we cannot define the temperature, pressure, etc. of the system and apply the thermodynamic math.

Consider system is compressed by a piston. If we compress it very slowly, the system is always in equilibrium from to .

For a particular state , the change of energy is given by

And in the ensemble average, we have

Therefore we derive that

If is not fixed but is fixed, we have