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Statistical Physics 2016-2017 Course Notes

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Statistical Physics (D20)

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Part II — Statistical Physics

Based on lectures by H. S. Reall

Notes taken by Dexter Chua

Lent 2017

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

Part IB Quantum Mechanics and “Multiparticle Systems” from Part II Principles of Quantum Mechanics are essential

Fundamentals of statistical mechanics Microcanonical ensemble. Entropy, temperature and pressure. Laws of thermody- namics. Example of paramagnetism. Boltzmann distribution and canonical ensemble. Partition function. Free energy. Specific heats. Chemical Potential. Grand Canonical Ensemble. [5]

Classical gases Density of states and the classical limit. Ideal gas. Maxwelldistribution. Equipartition of energy. Diatomic gas. Interacting gases. Virial expansion. Van der Waal’s equation of state. Basic kinetic theory. [3]

Quantum gases Density of states. Planck distribution and black body radiation. Debye model of phonons in solids. Bose–Einstein distribution. Ideal Bose gas and Bose–Einstein condensation. Fermi-Dirac distribution. Ideal Fermi gas. Pauliparamagnetism. [8]

Thermodynamics Thermodynamic temperature scale. Heat and work. Carnot cycle. Applications of laws of thermodynamics. Thermodynamic potentials. Maxwell relations. [4]

Phase transitions Liquid-gas transitions. Critical point and critical exponents. Ising model. Mean field theory. First and second order phase transitions. Symmetries andorder parameters. [4]

Contents II Statistical Physics

0 Introduction Contents
1 Fundamentals of statistical mechanics
- 1 Microcanonical ensemble
- 1 Pressure, volume and the first law of thermodynamics
- 1 The canonical ensemble
- 1 Helmholtz free energy
- 1 The chemical potential and the grand canonical ensemble
- 1 Extensive and intensive properties
2 Classical gases
- 2 The classical partition function
- 2 Monoatomic ideal gas
- 2 Maxwell distribution
- 2 Diatomic gases
- 2 Interacting gases
3 Quantum gases
- 3 Density of states
- 3 Black-body radiation
- 3 Phonons and the Debye model
- 3 Quantum ideal gas
- 3 Bosons
- 3 Bose–Einstein condensation
- 3 Fermions
- 3 Pauli paramagnetism
4 Classical thermodynamics
- 4 Zeroth and first law
- 4 The second law
- 4 Carnot cycles
- 4 Entropy
- 4 Thermodynamic potentials
- 4 Third law of thermodynamics
5 Phase transitions
- 5 Liquid-gas transition
- 5 Critical point and critical exponents
- 5 The Ising model
- 5 Landau theory
Index

1 Fundamentals of statistical mechanics

1 Microcanonical ensemble

We begin by considering a rather general system. Suppose we have an isolated system containingNparticles, whereNis a Large NumberTM. The canonical example to keep in mind is a box of gas detached from reality.

Definition(Microstate).Themicrostateof a system is the actual (quantum) state of the system. This gives a complete description of the system.

As one would expect, the microstate is very complicated and infeasibleto describe, especially when we havemany particles. In statistical physics, we observe that many microstates are indistinguishable macroscopically. Thus, we only take note of some macroscopically interesting quantities, and use these macroscopic quantities to put a probability distribution on the microstates. More precisely, we let{|n〉}be a basis of normalized eigenstates, say

Hˆ|n〉=En|n〉.

We let p(n) be the probability that the microstate is|n〉. Note that this probability isnotthe quantum probability we are used to. It is some probability assigned to reflect our ignorance of the system. Given such probabilities, we can define the expectation of an operator in the least imaginative way:

Definition(Expectation value).Given a probability distributionp(n) on the states, the expectation value of an operatorOis

〈O〉=

∑

p(n)〈n|O|n〉.

If one knows about density operators, we can describe the system as a mixed state with density operator

ρ=

∑

p(n)|n〉〈n|.

There is an equivalent way of looking at this. We can consider anensemble consisting ofW≫1 independent copies of our system such thatWp(n) many copies are in the microstate|n〉. Then the expectation is just the average over the ensemble. For most purposes, how we think about this doesn’t really matter. We shall further assume our system is inequilibrium, i. the probability distributionp(n) does not change in time. So in particular〈O〉is independent of time. Of course, this does not mean the particles stop moving. The particles are still whizzing around. It’s just that the statistical distribution does not change. In this course, we will mostly be talking about equilibrium systems. When we get out of equilibrium, things become very complicated. The idea of statistical physics is that we have some partial knowledgeabout the system. For example, we might know its total energy. The microstates that are compatible with this partial knowledge are calledaccessible. Thefundamental assumption of statistical mechanicsis then

An isolated system in equilibrium is equally likely to be in any of the accessible microstates.

Thus, different probability distributions, or different ensembles, are distinguished by the partial knowledge we know.

Definition(Microcanonical ensemble).In amicrocanonical ensemble, we know the energy is betweenEandE+δE, whereδEis the accuracy of our measuring device. The accessible microstates are those with energyE≤En≤E+δE. We let Ω(E) be the number of such states.

In practice,δEis much much larger than the spacing of energy levels, and so Ω(E)≫1. A priori, it seems like our theory will depend on what the value of δEis, but as we develop the theory, we will see that this doesn’t really matter. It is crucial here that we are working with a quantum system, so the possible states is discrete, and it makes sense to count the number of systems. We need to do more quite a bit work if we want to do this classically.

Example we haveN= 10 23 particles, and each particle can occupy two states|↑〉and|↓〉, which have the same energyε. Then we always haveNε total energy, and we have Ω(Nε) = 2 10

23 .

This is a fantastically huge, mind-boggling number. This is the kindof number we are talking about.

By the fundamental assumption, we can write

p(n) =

{ 1

Ω(E) ifE≤En≤E+δE 0 otherwise

.

This is the characteristic distribution of the microcanonical ensemble. It turns out it is not very convenient to work with Ω(E). In particular, Ω(E) is not linear inN, the number of particles. Instead, it scales as an exponential ofN. So we take the logarithm.

Definition(Boltzmann entropy).The(Boltzmann) entropyis defined as

S(E) =klog Ω(E),

wherek= 1. 381 × 10 − 23 J K− 1 isBoltzmann’s constant.

This annoying constantkis necessary because when people started doing thermodynamics, they didn’t know about statistical physics, and picked weird conventions. We wrote our expressions asS(E), instead ofS(E,δE). As promised, the value ofδEdoesn’t really matter. We know that Ω(E) will scale approximately linearly withδE. So if we, say, doubleδE, thenS(E) will increase byklog2, which is incredibly tiny compared toS(E) =klogΩ(E). So it doesn’t matter which value ofδEwe pick. Even if you are not so convinced that multiplying 10 10

23 by a factor of 2 or addinglog2 to 10 23 do not really matter, you should be reassured that at the end, we will rarely talk about Ω(E) orS(E) itself. Instead, we will often divide two different Ω’s to get probabilities, or differentiateSto get other interesting quantities. In these cases, the factors really do not matter.

Again recall that the numbers at stake are unimaginably huge. So ifS 1 (E(1)) + S 2 (E(2)) is even slightly different fromS(Etotal), then the probability is effectively zero. And by above, for the two quantities to be close, we needE(1)=E∗. So for all practical purposes, the value ofE(1)is fixed intoE∗. Now imagine we prepare two systems separately with energiesE(1)andE(2) such thatE(1) 6 =E∗, and then bring the system together, then we are no longer in equilibrium(1)will change until it takes valueE∗, and then entropy of the system will increase fromS 1 (E(1)) +S 2 (E(2)) toS 1 (E∗) +S 2 (Etotal−E∗). In particular, the entropy increases.

Law (Second law of thermodynamics). The entropy of an isolated system increases (or remains the same) in any physical process. In equilibrium, the entropy attains its maximum value.

This prediction is verified by virtually all observations of physics. While our derivation did not show it isimpossibleto violate the second law of thermodynamics, it is very very very very very very very veryunlikely to be violated.

Temperature

Having defined entropy, the next interesting thing we can define is thetemperature. We assume thatSis a smooth function inE. Then we can define the temperature as follows:

Definition(Temperature).Thetemperatureis defined to be

1 T

=

dS dE

.

Why do we call this the temperature? Over the course, we will see that this quantity we decide to call “temperature” does behave as we would expect temperature to behave. It is difficult to give further justification of this definition, because even though we vaguely have some idea what temperature is likein daily life, those ideas are very far from anything we can concretely write down or even describe. One reassuring property we can prove is the following:

Proposition interacting systems in equilibrium have the same tempera- ture.

Proof that the equilibrium energyE∗is found by maximizing

S 1 (Ei) +S 2 (Etotal−Ei)

over all possibleEi. Thus, at an equilibrium, the derivative of this expression has to vanish, and the derivative is exactly

dS 1 dE

∣

E(1)=E∗

−

dSi dE

∣

E(2)=Etotal−E∗

= 0

So we need 1 T 1

=

1 T 2

.

In other words, we need T 1 =T 2.

Now suppose initially, our systems have different temperature. Wewould expect energy to flow from the hotter system to the cooler system. This is indeed the case.

Proposition two systems with initial energiesE(1),E(2)and temper- aturesT 1 ,T 2 are put into contact. IfT 1 &gt; T 2 , then energy will flow form the first system to the second.

Proof we are not in equilibrium, there must be some energy transferfrom one system to the other. Suppose after timeδt, the energy changes by

E(1)7→E(1)+δE E(2)7→E(2)−δE,

keeping the total energy constant. Then the change in entropy is given by

δS=

dS 1 dE

δE(1)+

dS 2 dE

δE(2)=

(

1 T 1

−

1 T 2

)

δE.

By assumption, we know 1 T 1

−

1 T 2

&lt; 0 ,

but by the second law of thermodynamics, we knowδSmust increase. So we must haveδE &lt;0, i. energy flows from the first system to the second.

So this notion of temperature agrees with the basic properties of temperature we expect. Note that these properties we’ve derived only depends on the fact that 1 T is a monotonically decreasing function ofT. In principle, we could have picked anymonotonically decreasing function ofT, and set it toddES. We will later see that this definition will agree with the other definitions of temperature we have previously seen, e. via the ideal gas law, and so this is indeed the“right” one.

Heat capacity

As we will keep on doing later, we can take different derivatives to get different interesting quantities. This time, we are going to get heat capacity that Twas a function of energy,T=T(E). We will assume that we can invert this function, at least locally, to getEas a function ofT.

Definition(Heat capacity).Theheat capacityof a system is

C=

dE dT

.

Thespecific heat capacityis

C mass of system

.

and S(E) =klog

(

N!

N↑!(N−N↑)!

)

.

This is not an incredibly useful formula. Since we assumed thatNandN↑are huge, we can useStirling’s approximation

N! =

√

2 πNNNe−N

(

1 +O

(

1 N

))

.

Then we have

logN! =NlogN−N+

1

2

log(2πN) +O

(

1 N

)

.

We just use the approximation three times to get

S(E) =k(NlogN−N−N↑logN↑+N↑−(N−N↑) log(N−N↑) +N−N↑)

=−k

(

(N−N↑) log

(

N−N↑

N

)

+N↑log

(

N↑

N

))

=−kN

((

1 −

E

Nε

)

log

(

1 −

E

Nε

)

+

E

Nε

log

(

E

Nε

))

.

This is better, but we can get much more out of it if we plot it:

E

S(E)

0

Nε/ 2 Nε

Nklog 2

The temperature is 1 T

=

dS dT

=

k ε log

(

Nε E

− 1

)

,

and we can invert to get

N↑ N

=

E

Nε

=

1

eε/kT+ 1

.

Suppose we get to control the temperature of the system, e. if we putit with a heat reservoir. What happens as we vary our temperature?

AsT→0, we haveN↑→0. So the states all try to go to the ground state.
AsT→∞, we findN↑/N→ 12 , andE→Nε/2.

The second result is a bit weird. AsT→∞, we might expect all things to go the maximum energy level, and not just half of them. To confuse ourselves further, we can plot another graph, forT 1 vsE. The graph looks like

E

1 T

0

Nε/ 2 Nε

We see that having energy&gt; Nε/2 corresponds to negative temperature, and to go from positive temperature to negative temperature, we need to passthrough infinite temperature. So in some sense, negative temperature is “hotter” than infinite temperature. What is going on? By definition, negativeTmeans Ω(E) is a decreasing function of energy. This is a very unusual situation. In this system, all the particles are fixed, and have no kinetic energy. Consequently, thepossible energy levels are bounded. If we included kinetic energy into the system, then kinetic energy can be arbitrarily large. In this case, Ω(E) is usually an increasing function ofE. NegativeThas indeed been observed experimentally. This requires setups where the kinetic energy is not so important in the range of energies weare talking about. One particular scenario where this is observed is in nuclear spins of crystals in magnetic fields. If we have a magnetic field, then naturally, most of the spins will align with the field. We now suddenly flip the field,and then most of the spins are anti-aligned, and this can give us a negative temperaturestate. Now we can’t measure negative temperature by sticking a thermometerinto the material and getting a negative answer. Something thatcanbe interestingly measured is the heat capacity

C=

dE dT

=

Nε 2 kT 2

eε/kT (eε/kT+ 1) 2

.

This again exhibits some peculiar properties. We begin by looking at a plot:

1 Pressure, volume and the first law of thermodynamics

So far, our system only had one single parameter — the energy. Usually, our systems have other external parameters which can be varied. Recall that our “standard” model of a statistical system is a box of gas. If we allow ourselves to move the walls of the box, then the volume of the system may vary. Aswe change the volume, the allowed energies eigenstates will change. So now Ω, and henceSare functions of energyandvolume:

S(E,V) =klog Ω(E,V).

We now need to modify our definition of temperature to account for thisdepen- dence:

Definition(Temperature).Thetemperatureof a system with variable volume is 1 T

=

(

∂S

∂E

)

,

withVfixed.

But now we can define a different thermodynamic quantity by taking the derivative with respect toV.

Definition(Pressure).We define thepressureof a system with variable volume to be p=T

(

∂S

∂V

)

.

Is this thing we call the “pressure” any thing like what we used to think of as pressure, namely force per unit area? We will soon see that this is indeed the case. We begin by deducing some familiar properties of pressure.

Proposition as before two interacting systems where the total volume V=V 1 +V 2 is fixed by the individual volumes can vary. Then the entropy of the combined system is maximized whenT 1 =T 2 andp 1 =p 2.

Proof have previously seen that we needT 1 =T 2. We also want ( dS dV

)

= 0.

So we need ( dS 1 dV

)

=

(

dS 2 dV

)

.

Since the temperatures are equal, we know that we also needp 1 =p 2.

For a single system, we can use the chain rule to write

dS=

(

∂S

∂E

)

dE+

(

∂S

∂V

)

dV.

Then we can use the definitions of temperature and pressure to write

Proposition(First law of thermodynamics).

dE=TdS−pdV.

This law relates two infinitesimally close equilibrium states. This is sometimes called thefundamental thermodynamics relation. Example a box with one side a movable piston of areaA. We apply a forceFto keep the piston in place.

⇐=F

What happens if we move the piston for a little bit? If we move througha distance dx, then the volume of the gas has increased byAdx. We assumeSis constant. Then the first law tells us

dE=−pAdx.

This formula should be very familiar to us. This is just the work doneby the force, and this must beF=pA. So our definition of pressure in terms of partial derivatives reproduces the mechanics definition of force per unitarea. One has to be cautious here. It is not always true that−pdVcan be equated with the word done on a system. For this to be true, we require the change to bereversible, which is a notion we will study more in depth later. For example, this is not true when there is friction. In the case of a reversible change, if we equate−pdVwith the work done, then there is only one possible thingTdScan be — it is the heat supplied to the system. It is important to remember that the first law holds foranychange. It’s just that this interpretation does not. Example the irreversible change, where we have a “free expansion” of gas into vacuum. We have a box

gas vacuum

We have a valve in the partition, and as soon as we open up the valve, the gas flows to the other side of the box. In this system, no energy has been supplied. So dE= 0. However, dV &gt;0, as volume clearly increased. But there is no work done on or by the gas. So in this case,−pdV is certainly not the work done. Using the first law, we know that TdS=pdV. So as the volume increases, the entropy increases as well.

where we define

Notation(β). β=

1 .

Note that while we derived this this formula under the assumption thatEn is small, it is effectively still valid whenEnis large, because both sides are very tiny, and even if they are very tiny in different ways, it doesn’tmatter when we add over all states. Now we can write thetotalnumber of microstates ofS+Ras

Ω(Etotal) =

∑

ΩR(Etotal−En) =ek

− 1 SR(Etotal)∑ n

e−βEn.

Note that we are summing over all states, not energy. We now use the fundamental assumption of statistical mechanics that all states ofS+Rare equally likely. Then we know the probability thatSis in state|n〉is

p(n) =

ΩR(Etotal−En) Ω(Etotal)

=

e−βEn ∑ ke −βEk.

This is called theBoltzmann distributionfor the canonical ensemble. Note that at the end, all the details have dropped out apart form the temperature describes the energy distribution of a system with fixed temperature. Note that ifEn≫kT= 1 β, then the exponential is small. So only states withEn∼kThave significant probability. In particular, asT→0, we have β→∞, and so only the ground state can be occupied. We now define an important quantity.

Definition(Partition function).Thepartition functionis

∑

e−βEn.

It turns out most of the interesting things we are interested in can be expressed in terms ofZand its derivatives. Thus, to understand a general system, what we will do is to compute the partition function and express it in somefamiliar form. Then we can use standard calculus to obtain quantities we are interested in. To begin with, we have p(n) = e−βEn Z

.

Proposition two non-interacting systems, we haveZ(β) =Z 1 (β)Z 2 (β).

Proof the systems are not interacting, we have

Z=

∑

n,m

e−β(E

(1)n+En(2)) =

(

∑

e−βE n(1)

)(

∑

e−βE

(2)n

)

=Z 1 Z 2.

Note that in general, energy isnotfixed, but we can compute the average value: 〈E〉=

∑

p(n)En=

∑Ene−βEn Z

=−

∂

∂β

logZ.

This partial derivative is taken with allEifixed. Of course, in the real world, we don’t get to directly change the energy eigenstates and see what happens. However, they do depend on some “external” parameters, such as the volumeV, the magnetic fieldBetc. So when we take this derivative, we have to keep all those parameters fixed. We look at the simple case whereVis the only parameter we can vary. Then Z=Z(β,V). We can rewrite the previous formula as

〈E〉=−

(

∂

∂β

logZ

)

.

This gives us the average, but we also want to know the variance ofE. We have

∆E 2 =〈(E−〈E〉) 2 〉=〈E 2 〉−〈E〉 2.

On the first example sheet, we calculate that this is in fact

∆E 2 =

(

∂ 2

∂β 2 logZ

)

=−

(

∂〈E〉

∂β

)

.

We can now convertβ-derivatives toT-derivatives using the chain rule. Then we get

∆E 2 =kT 2

(

∂〈E〉

∂T

)

=kT 2 CV.

From this, we can learn something important. We would expect〈E〉∼N, the number of particles of the system. But we also knowCV∼N. So ∆E 〈E〉

∼

1 √

N

.

Therefore, the fluctuations are negligible ifNis large enough. This is called the thermodynamic limitN→∞. In this limit, we can ignore the fluctuations in energy. So we expect the microcanonical ensemble and the canonical ensemble to give the same result. And for all practical purposes,N∼ 1023 is a large number. Because of that, we are often going to just writeEinstead of〈E〉.

Example we had particles with

E↑=ε, E↓= 0.

So for one particle, we have

Z 1 =

∑

e−βEn= 1 +e−βε= 2e−βε/ 2 cosh

βε 2

.

If we haveN non-interacting systems, then since the partition function is multiplicative, we have

Z=Z 1 N= 2ne−βεN/ 2 coshN

βε 2

.

Example the microcanonical ensemble, we have

p(n) =

{

1 Ω(E) E≤En≤E+δE 0 otherwise

Then we have

S=−k

∑

n:E≤En≤E+δE

1 Ω(E)

log

1 Ω(E)

=−kΩ(E)·

1 Ω(E)

log

1 Ω(E)

=klog Ω(E).

So the Gibbs entropy reduces to the Boltzmann entropy.

How about the canonical ensemble?

Example the canonical ensemble, we have

p(n) =

e−βEn Z

.

Plugging this into the definition, we find that

S=−k

∑

p(n) log

(

e−βEn Z

)

=−k

∑

p(n)(−βEn−logZ)

=kβ〈E〉+klogZ,

using the fact that

∑

p(n) = 1. Using the formula of the expected energy, we find that this is in fact

S=k

∂

∂T

(TlogZ)V.

So again, if we want to compute the entropy, it suffices to find a nice closed form ofZ.

Maximizing entropy It turns out we can reach the canonical ensemble in a different way. The second law of thermodynamics suggests we should always seek to maximize entropy. Now if we take the optimization problem of “maximizing entropy”, what probability distribution will we end up with? The answer depends on what constraints we put on the optimization problem. We can try to maximizeSGibbsover all probability distributions such that p∑(n) = 0 unlessE≤En≤E+δE. Of course, we also have the constraint p(n) = 1. Then we can use a Lagrange multiplierαand extremize

k− 1 SGibbs+α

(

∑

p(n)− 1

)

,

Differentiating with respect top(n) and solving, we get

p(n) =eα− 1.

In particular, this is independent ofn. So all microstates with energy in this range are equally likely, and this gives the microcanonical ensemble. What about the canonical ensemble? It turns out this is obtained by max- imizing the entropy over allp(n) such that〈E〉is fixed. The computation is equally straightforward, and is done on the first example sheet.

1 Helmholtz free energy

In the microcanonical ensemble, we discussed the second law of thermodynamics, namely the entropy increases with time and the maximum is achieved in an equilibrium. But this is no longer true in the case of the canonical ensemble, because we now want to maximize the total entropy of the system plus the heat bath, instead of just the system itself. Then is there a proper analogous quantity for the canonical ensemble? The answer is given by the Helmholtz free energy.

Definition(Helmholtz free energy).TheHelmholtz free energyis

F=〈E〉−TS.

As before, we will often drop the〈·〉. In general, in an isolated system,Sincreases, andSis maximized in equilib- rium. In a system with a reservoir,Fdecreases, andFminimizes in equilibrium. In some senseFcaptures the competition between entropy and energy. Now is there anything analogous to the first law

dE=TdS−pdV?

Using this, we can write

dF= dE−d(TS) =−SdT−pdV.

When we wrote down the original first law, we had dSand dVon the right, and thus it is natural to consider energy as a function of entropy and volume (instead of pressure and temperature). Similarly, It is natural to think ofF as a function ofTandV. Mathematically, the relation betweenFandEis thatF is theLegendre transformofE. From this expression, we can immediately write down

S=−

(

∂F

∂T

)

,

and the pressure is p=−

(

∂F

∂V

)

.

As always, we can express the free energy in terms of the partition function.

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Statistical Physics 2016-2017 Course Notes

Module: Statistical Physics (D20)

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Part II — Statistical Physics

Based on lectures by H. S. Reall

Notes taken by Dexter Chua

Lent 2017

These notes are not endorsed by the lecturers, and I have modified them (often

significantly) after lectures. They are nowhere near accurate representations of what

was actually lectured, and in particular, all errors are almost surely mine.

Part IB Quantum Mechanics and “Multiparticle Systems” from Part II Principles of

Quantum Mechanics are essential

Fundamentals of statistical mechanics

Microcanonical ensemble. Entropy, temperature and pressure. Laws of thermody-

namics. Example of paramagnetism. Boltzmann distribution and canonical ensemble.

Partition function. Free energy. Specific heats. Chemical Potential. Grand Canonical

Ensemble. [5]

Classical gases

Density of states and the classical limit. Ideal gas. Maxwell distribution. Equipartition

of energy. Diatomic gas. Interacting gases. Virial expansion. Van der Waal’s equation

of state. Basic kinetic theory. [3]

Quantum gases

Density of states. Planck distribution and black body radiation. Debye model of

phonons in solids. Bose–Einstein distribution. Ideal Bose gas and Bose–Einstein

condensation. Fermi-Dirac distribution. Ideal Fermi gas. Pauli paramagnetism. [8]

Thermodynamics

Thermodynamic temperature scale. Heat and work. Carnot cycle. Applications of laws

of thermodynamics. Thermodynamic potentials. Maxwell relations. [4]

Phase transitions

Liquid-gas transitions. Critical point and critical exponents. Ising model. Mean field

theory. First and second order phase transitions. Symmetries and order parameters. [4]