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• Thermodynamics, Statistical Mechanics and LargeDeviations

Stefano OllaCourse Notes

November 2011

• 2

• Chapter 1

Thermodynamics: a crash course

1.1 Thermodynamic equilibrium states

From a mechanical point of view, the equilibrium state of an elastic wire is char-acterized by its length L, that can be changed applying a tension (force) on theextremes. The resulting length is an increasing function of : L = F (). We alsoobserve that this function F depends also on the temperature of the wire, typicallyincreasing with . The first object of thermodynamics is to introduce this parameter, whose definition (or measurement) is much more delicate than L or .

The definition of temperature goes through defining when two systems are atthe same temperature, by what is called the 0th principle of thermodynamics : 1:

If a system A remains in equilibrium when isolated and placed inthermal contact first with system B and then with system C, the equi-librium of B and C will not be disturbed when they are placed in contactwith each other.

Here remains in equilibrium means that the relation between L and doesnot change. We also use here the concept of isolated system and thermal contact,that require respectively the notion of adiabatic wall and conductive wall.

The system A and B are separated by an adiabatic wall if they can havedifferent equilibrium relation between L and . They are separated by a conductivewall if the must have the same equilibrium relation.

1The numbering of the principles in thermodynamics follows a inverse chronological order: thesecond principle was postulated by Carnot in 1824, the first principle was clearly formulated byHelmholtz and Thomson (Lord Kelvin) in 1848, while the need of the zero principle announcedhere was realized by Fowler in 1931. See the detailed discussion in the first chapter of Zemansky

3

• 4 CHAPTER 1. THERMODYNAMICS: A CRASH COURSE

One could see all these as circular definition, in fact all this is equivalent aspostulating the existence of adiabatic and diathermic (thermally conductive) wallsthat are defined as devices that have the above properties. From all this we obtainthe existence of the parameter that we call temperature (see in Zemanski a verydetailed discussion of this point).

So we can define the equilibrium relation L = L(, ). Since it is strictlyincreasing in both variable we can also write = (L, ), as well as = (L, ), i.e.any two of these three variables can be chosen independently in order to characterizea thermodynamic equilibrium state.

1.2 Differential changes of equilibrium states

Suppose we have our wire in the thermodynamic equilibrium defined by the value, . This can be obtained by applying a tension to one extreme and fixing theother to a wall, and applying a thermal bath at temperature , for example a verylarge (infinite) system at this temperature on the other side of the conductive wall.If we perform infinitesimal changes of these parameters, they imply an infinitesimalvariation dL of the lenght:

dL =(L

)

d +

(L

)

d (1.2.1)

These partial derivatives are connected with physical important quantities that canbe measured experimentally:

the linear dilation coefficient:

=1

L

(L

)

(1.2.2)

Experimentally it is observed that (, ) depends little by , but changes verystrongly with .

the isothermal Young modulus

Y =LA

(

L

)

(1.2.3)

where A is the section of the wire. Experimentally Y depends little on andstrongly on .

We also call C =1L

(L

)

the isothermal compressibility.

• 1.2. DIFFERENTIAL CHANGES OF EQUILIBRIUM STATES 5

It is an elementary exercise to prove that(

L

)

(L

)

= (

)L

(1.2.4)

and consequently (

)L

= C

(1.2.5)

An infinitesimal variation of the tension can be written in function of d and dL:

d =

(

)Ld +

(

L

)

dL = Cd +

1

CLdL (1.2.6)

At constant volume we have

d = Cd (1.2.7)

One of the main issues in discussing foundations of thermodynamics is thephysical meaning of these differential changes of equilibrium states. In principle,as we actually change the tension of the cable, the system will go into a sequenceof non-equilibrium states before to relax to the new equilibrium. But, quotingZemanski, thermodynamics does not attempt to deal with any problem involving therate at which the process takes place. And, always quoting Zemanski:

Every infinitesimal in thermodynamics must satisfy the requirementthat it represents a change in a quantity which is small with respect tothe quantity itself and large in comparison with the effect produced bythe behavior of few molecules.

1.2.1 Work

In an arbitrary quasi static infinitesimal transformation, the differential form dL iscalled work (or differential work). It is clear that this is not an exact differential form,but in thermodynamics books it is used the notation d\W . This is elementary, lookingat the path of a transformation in the (,L) coordinates frame. In performing aclosed path, that we call cycle, maybe through a sequence of isobar and isocoretransformations, the path integral

d\W 6= 0 (equal to the area inside the path),

and represent the work done on the system by the external force (tension)2.

2Mathematically the difference with the notion of work in mechanics is that here the force isalso a function of the temperature

• 6 CHAPTER 1. THERMODYNAMICS: A CRASH COURSE

1.2.2 Internal Energy and Heat exchange

The point of the first principle of thermodynamics is to preserve the mechan-ical notion of energy as a conserved quantity also during thermodynamic transfor-mations, which means to assume the existence of a state function U(L, ) thatrepresent the internal energy of the system in the corresponding thermodynamicequilibrium state. During a quasi-static thermodynamic infinitesimal transforma-tion, this energy is modified by the work d\W and, since dU has to be an exactdifferential, by some other (not exact) differential form d\Q called heat exchange:

dU = dL+ d\Q (1.2.8)

It is important to notice that work and heat are determined by specifying the processof change, and they are not functions of the state of the system. Mathematically thismeans they are not exact differentials. As we have already said, in mechanics anychange of the energy of a system is caused by the work done by external forces. If wewant to reduce the first principle to a purely mechanical interpretation (that will bethe scope of statistical mechanics), this will be the following. The system has many(a very large number) degrees of freedom and many external forces acting on them.Some, few, of these forces are controlled, ordered, macroscopic and slow, and thework done by these we still call it work, in our case is this ordered and controlledslow force, and dL the work associated. The other forces are many, uncontrolled(or disordered, in the sense that we do not have information on them), microscopicand fast. The amount of this uncontrolled or disordered work or exchange of energywe call heat.

One of the main problem of the statistical mechanic interpretation of thermo-dynamics is to separate the slow macroscopic degree of freedom that generate workfrom the fast microscopic ones that generate heat. The slow degrees of freedom aregenerally associated to conserved quantities of the isolated system (with no externalforces acting on it or thermal contact with other systems).

So the first principle (1.2.8) defines this separation of scales (in space and intime), whose mechanical explanation impose the use of probability to describe theuncontrolled forces.

1.2.3 Thermodynamic transformations and cycles

We can represent a finite thermodynamic transformation by integration along pathof the differential forms defined above. Each choice of a path defines a differentthermodynamic process or quasi static transformation. Depending on the type oftransformation it may be interesting to make a different choice of the coordinatesto represent it graphically.

• 1.2. DIFFERENTIAL CHANGES OF EQUILIBRIUM STATES 7

Often is used the L diagrams.

The first diagram on the left describe a quasi-static transformation for lenghtLi to Lf . If this is happening for example as a free expansion means that the tension is decreasing, but it could be increasing if instead is pulling with respect themechanical equilibrium. The second diagram represent a compression from Lf toLi, and the third a so called cycle, returning to the original state. The shaded arearepresent the work done during the transformation (taken with the negative sign inthe second diagram). In the third the work is given by the integral along the cycle

W =

dL (1.2.9)

that by the first principle will be equal to Q, where Q is the total heat producedby the process during the cycle and transmitted to the exterior (or absorbed by theexterior, depending from the sign).

There are some important thermodynamic quasi static transformation we wantto consider:

Isothermal transformations : While a force perform work on the system, thisis in contact with a thermostat, a huge system in equilibrium at a giventemperature , so big that the exchange of heat with our elastic does notperturb the equilibrium state of the thermostat. Ideally a thermostat is aninfinite system. During a isothermal transformation only the length L changesas effect of the change of the tension d , and the infinitesimal exchanges ofheat and work are related by

d\W = dL = (L

)

d = d\Q+ dU (1.2.10)

The isothermal tra