Entropy and temperature

Fundamental assumption: an isolated system (N, V and U and all external parameters constant) is equally likely to be in any of the quantum states accessible to it!

A quantum state is accessible if its properties are compatible with the physical specification of the system

Probability of a state s: P(s). Let g be the number of accessible states --> P(s)=1/g if the state s is accessible, and P(s)=0 otherwise

Properties: ;1)(}{

s

sP }{ }{

1)()()(

s s gsxsPsxx

Ensemble: a collection of many “copies” (replica) of the original system. If there are g accessible states there are g replicas, one in each accessible quantum state. Averages calculated by us are all ensemble averages

Ergodic hypothesis: the thermodynamic systems goes over all accessible states in a very short time (in a time much shorter than needed for the measurement of a macroscopic parameter)

Basic hypothesis for an isolated system: time average equals the ensemble average

Example: Ensemble of a system of 10 spins with energy -8mB and spin exces 2s=8.

Most probable configuration:

- Let two systems (S1 and S2 be in contact) so that energy can be transferred from one to the other (thermal contact)

- S=S1+S2 another larger system. We have for internal energy: U=U1+U2.

- let us consider S isolated (U, V, N….constant)

- what determines whether there will be a net flow of energy from one system to another? --> concept of temperature- the most probable division of the total energy is that for which the combined system has the maximum number of accessible states!

- let us consider the case of thermal contact of two systems. The numbers of particles in system 1 is N1 and in system 2 is N2, the values of the

energies in system 1 is U1 and in system 2 U2. The total energy U=U1+U2

is fixed.),(),(),(

1

122111 s

UUNgUNgUNg

(g(N,U) is the multiplicity function of system S, g1,2 for system 1,2 respectively.)

a configuration : set of all states with specified values of U1 and U2..

most probable configuration: the configuration for which g1g2 is maximum

- if N>>1 fluctuations about the most probable configurations are small

Schematic representation of the dependence of the configuration multiplicity on the division of the total energy between the systems S1 and S2

- values of the average physical properties of a large system in thermal contact with another large system are accurately described by the most probable configuration! --> such average values are called thermal equilibrium values

average of a physical quantity over all accessible states --> average over the most probable configuration

let us consider the case of thermal contact of two spin systems in magnetic field. The numbers of spins in system 1 is N1 and in system 2 is N2, the values of the spin excess 2s1, 2s2 may be different and can change but the total energy: U=U1(s1)+U2(s2)=-2mB(s1+s2)=-2mBs is fixed.

N

Ns

N

s

N

s

s

eggssNgsNgsNg2/1

2/1

)22

(

21122111

1

2

21

1

21

1

)0()0(),(),(),(

(g(N,s) is the multiplicity function of system S, g1,2 for system 1,2 respectively. If N1<N2 then in the sum -1/2N1 s11/2N1)

Example: Two spin systems in thermal contact

the most probable configuration of the system is where:

0)]ˆ,(ln[)]ˆ,(ln[ 2221111

sNgsNgs N

s

N

s

N

ss

N

s

2

2

2

1

1

1 ˆˆˆ

Nearly all accessible states of the combined system satisfy or nearly satisfy (*)!

(*)

N

s

egggg

22

21max21 )0()0()(

sharpness of g1g2:

22

11

ˆ

ˆ

ss

ss2

2

1

2 22

max21222111 )(),(),( NNeggsNgsNg

Numerical example: N1=N2=1022; =1012; /N1=10-10 --> 2 2/N1=200 --> g1g2 reduced to e-40010-174 of its maximal value!

We may expect to observe substantial fractional deviations from the most probable configuration only for a small system in thermal contact with a large system!

Thermal equilibrium

consider the case of thermal contact of two systems. The numbers of particles in system 1 is N1 and in system 2 is N2,

the values of the energies in system 1 is U1 and in system 2 U2. The total energy U=U1+U2 is fixed.

),(),(),(1

122111 s

UUNgUNgUNg (g(N,U) is the multiplicity function of system S, g1,2 for system 1,2 respectively.)

in thermal equilibrium the largest term in the above sum governs the properties of the total system!

in thermal equilibrium:

0

0

21

212

212

1

1

21

dUdU

dUgU

gdUg

U

gdg

NN

212

2

1

1 )ln()ln(

NNU

g

U

g

we define a quantity , called entropy )],(ln[),( UNgUN

In thermal equilibrium we have:

2121 NN

UU

The concept of temperature:

- in thermal equilibrium the temperatures of the two systems are equal! 21 TT one can define the absolute temperature as:

NB U

kT

1

kB is the Boltzmann constant, kB=1.381 x 10-23 J/K

the fundamental temperature, : NU

1

TkB has the dimensions of energy

Temperature scales:

(defined experimentally by thermometers ):

we need two reproducible temperatures and divide he interval betweenthem in equal divisions

II. temperatureboiling point of pure waterI. temp.=32 0 F; II. temp.=212 0 F Fahrenheit temperature scale

Celsius temperature scaleI. temp.=0 0 C; II. temp.=100 0 C

Kelvin temperature scale or absolute temperature scaledefined by the properties of ideal gases, distance between units is kepttemperature the lowest possible temperature where the volume of idealgases would go to zero 0 0 K=-273 0 CII. temperature where water freezes 273 0 K=0 0 C

Problems:

1. Solve problem 1. (“Entropy and temperature” on page 52)

2. Solve problem 2. (“Paramagnetism” on page 52)

3. Calculate the multiplicity function for a harmonic oscillator (example on page 24)

4. Solve problem 3. (“Quantum harmonic oscillator” on page 52)

Extra problem:

Consider a binary magnetic system formed by 5 spins ( ) arranged in a row. The spins are

interacting only between themselves, each spin with it’s two (or one - for spin 1 and 5) nearest neighbor.

The interaction energy between two nearest neighbor spin i and k is: U(i,k)=-ik. . Let U be the total

energy of the system. Determine the g(U) multiplicity function for this system. (Write up the possible

values for U, and correspond g values)

1i