Chapter 4

Isolated Paramagnet – Subsystems and Temperature

4.1 Microscopic states and thermodynamic equilibrium

So far only one new concept has been introduced beyond the dynamic of the system, which is

the probabiity within a set of states, or an ensemble. It is time to try and connect this new

concept with thermodynamic quantities. The firs among these is the temperature. We turn

therefore to the identification of the relative temperature of two systems.

obviously, in order to discuss temperatur we need at least two systems, since the

temperature is precisely the intensive variable, whose equality characterizes the equilibrium

between them when there is no mechanical interaction. We choose, therefore, two

paramagnetic systems:

System α with Nα spin and a magnetic field Hα ,

System b with Nb spin and a magnetic field Hb .

We isolate both systems from the rest of the universe, but allow them to interact

thermally. This means that the total energy of the two systems E, will be constant but the

energies of the two systems Eα and Eb are unconstrained provided Eα + Eb = E. No forces will

act between the spins, except for the tiny forces we mentioned earlier which drive the system

towards thermal equilibrium – namely, to uniform occurrence rates of all microscopic states

of the combined isolated system, as explained in chap.3.

The energy of a given state of system α is

Eα = -μB Hα ∑i=1

N α

σ i (2.4.1)

and that of system b is

Eb = E - E α (2.4.2)

4.2 and the temperature

If the number of states of system α with energi Eα is (Eα), then system b has (E - Eα)

states, and the total number of states of the composite system, for which system α has energy

Eα is

T = (Eα , Hα , Nα) . (Eb , Hb , Nb)

= (Eα , Hα , Nα) . (E - Eα , Hb , Nb) (2.4.3)

Where we have emphasized the dependence of both factors on the number of spins of the

subsystems and on the magnetic field of each of them.

The rest of the argument has the following structure:

When the numbers Nα and Nb are very large, there exists a value of Eα, which we

denote by Eα, for which T is maxsimal. Moreover, the maximum is extremely sharp, and the

number of states in which Eα differs from Eα is, relatively, very small. This will be the

equilibrium state, because if the combined system “visits” all the states with total energy E at

the same frequency, it will almost always be in a state for which Eα = Eα. In this case we can

identify the intensive quantity that becomes equal in the two subsystems. This quantity will

be called the temperature.

In order to proced, we write (2.4.3) in the form

T = exp ( S (Eα , H α , N α )+S (E−Eα , H b ,N b )k ) (2.4.4)

Where for each system we have defined separately Sk = ln

The number of states attains its maximum at Eα , which may be determined by the

requirement that the derivative of the exponent with respect to Eα should vanish, or

1k ∂S (Eα ,H α ,N α )

∂E α|Eα =

1k ∂S (Eb, H b , N b )

∂ Eb|Eb ≡ , (2.4.5a)

Where Eb is not an independent variable but satisfies Eb = E - Eα and Eb = E - Eα and has the

dimensions of energy to the power of – 1.

Since is a monotonic function of S, if S has a maximum is maximal as well.

In conclusion, we found an intensive quantity which characterizes the maximum of T

and has the same value in the two subsystems. We called this quantity . Second, if we also

find that almost all the states of the composite system satisfy Eα = Eα, then this will be the

system’s thermal equilibrium state.

4.2 and the temperature

Before we proceed to prove the sharpness of the maximum, let us identify the quantity in

our model of a paramagnet. came about from equilibrium conciderations between two

systems α dan b, but it is possible to define it in general for a single paramagnetic system

with a given Energy E :

= 1k∂S (E ,H , N )

∂ E(2.4.5b)

The “entropy” S of an isolated paramagnet is given by Eq.(2.3.13) and by differentiating it be

obtain

= 1

2μ BH ln [( 12− E /N

2μB H )/( 12+ E/N

2μ B H )] (2.4.6)

Recalling the expression (2.3.7) and (2.3.8), for the probabilities for a spin to point up or

down, we can write

2μBH = ln [ P (σ=+1 )P (σ=−1 ) ] (2.4.7)

From which we obtain the interesting result

P (+1 )P (−1 )

= e2μBH (2.4.8)

Exercise 4.1

Use Eq. (2.4.8) to calculate P (+1 )and P (−1 ). Compare to (2.3.7) and (2.3.8).

Solution on page 201

Namely, if we knew that = 1kT , then (2.4.8) would be the expression for the

Boltzmann distribution, as in part I. That is, the ratio of the probabilities of the two states is

e−∆ E / kT, where ∆ E = is their energy difference.

But a present we canot conclude that 1/ is proportional to an absolute temperature, only that

it is an increasing function of the relative temperature – since even after demonstrating the

sharpness of the maximum, we will only know that is identical for systems at thermal

equilibrium with each other. In order to identify as defined by Eq. (2.4.5) as an absolute

temperature, we have to show that it connects the entropy change with the heat increase, or

that it may be identified from the ideal gas law. However, we may note that if S is indeed the

entropy, then (2.4.5b) is the connection between the entropy and the absolute temperature.

4.3 Sharpness of the maximum

In order to find the behavior of the number of states of the combined system as a function of

Eα near its maximum, we will use the expression for (E) when N ≫ 1, i.e. Eq. (2.3.14).

Inserting it into Eq.(2.4.3) and taking for simplicity Hα = Hb = H , we obtain

T = Cα Cb exp ( −Eα2

2N α μB2 H 2 ) exp [ – (E−Eα )2

2N α μB2 H2 ] , (2.4.9)

Where Cα and Cb are normalization constants that depend on Nα and Nb. Interms of the

“entropy” we obtain

1k ST =

1k (Sα+Sb ) = ln T = ln (Cα Cb) -

12 (μBH )2 [ Eα2N α

+(E−Eα )2

Nb ] (2.4.10)

Next we find the maximum of the entropy. Since (2.4.10) is a quadratic function of Eα , there

is no need to diferentiate with respect to Eα ; it is enough to complete the expression in

brackets to a square:

Eα2

N α+

(E−Eα )2

Nb=

(N ¿¿α+Nb)Eα2

NαN b¿ -

2E EαNb

+ E2

N b

= N α+N b

N αNb (Eα− Nα E

Nα+Nb )2

+ E2

N α+N b (2.4.11)

so that

1k ST = ln (Cα Cb) -

12 (μBH )2 [ N α+Nb

NαN b (Eα− Nα EN α+Nb )

2

+ E2

Nα+N b ] (2.4.12)

This is quadratic function of Eα whose maximum is attained at

Eα = N αEN α+N b

(2.4.13)

As will immediatelly be shown, T has a sharp maximum at this energy, so that this will also

be the average of Eα. We see therefore, that the energy is distributed between the two

systems in direct proportionality to their size. In order to show that the overwhelming

majority of states are concentrated around Eα = Eα , We return to T , and rewrite it using

(2.4.12) in the form

T(Eα) = CT exp [ −N α+N b

2 (μBH )2NαN b(E−Eα )2], (2.4.14)

Where the constant CT includes all the factors which do not depend on Eα.