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Study notes for Statistical Physics:
A concise, unified overview of the subject
70
Phase transitions Hence
E =−J
i,j
SiSj=−JzN 2M2,
where M = S ≡ order parameter. From the thermodynamic definition of the heat capacity, CB at constant magnetic field, we have
CB = ∂E
∂T
B
=−2JzN 2MdM
dT =−JzN MdM dT . Now, for:
T > Tc : M = 0 therefore CB = 0;
T ≤Tc : M = (−3θc)1/2.
Thus ∂M
∂T = 1
2(−3θc)−1/2× −dθc
dT = −3
2 M−1dθc
dt = −3
2 M−1Tc−1, and so
∂E
∂T
B
= 3
2JzN M M−1Tc−1= 3 2
JzN
c ,from equation (5.17).
= 3
2N k as Jz=kTc. HenceCB is discontinuous atT =Tc and soα= 0.
CASE 2: β
The mean magnetization M =S0, and from mean field theory:
S0= tanh(βB+ 2zJβS0).
Hence we can write:
M = tanh(βzJM+b) whereb≡βB.
Now mean field theory gives zβcJ= 1 orzJ = 1/βc, thus it follows that M = tanh
βM βc
+b
= tanh
MTc T +b
= tanh M
(1 +θc)+b
. Set B= 0 and expand forT ∼Tc, in which case θcis small:
M = M
1 +θc −1 3
M3 (1 +θc)3, and re-arranging:
M
1− 1 1 +θc
=−1 3
M3 (1 +θc)3, hence, either:
M = 0 or
M2=−3θc
(1 +θc)3
(1 +θc) =−3θc(1 +θc)2. Taking the nontrivial case,
M ∼ | −3θc|1/2,
and by comparison with the equation which defines the critical exponent:
β= 1/2.
52
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Study notes for Statistical Physics:
A concise, unified overview of the subject
71
Phase transitions
CASE 3: γ andδ From the definition of the isothermal susceptibility χT, we have:
χT = ∂M
∂B =β∂M
∂b , and also
M = tanh M
1 +θc +b
M
1 +θc+b for T > Tc. Now, with some re-arrangement,
M− M
1 +θc
=b, to this order of approximation and, re-arranging further, we have:
M =
1 +θc θc
b.
Hence
χT ∼ ∂M
∂b ∼ 1 θc
as θc→0 and so
χT ∼θ−1c , γ=−1,
which follows from the definition of γ. Next, consider the effect of an externally imposed field at T = Tc, whereθc= 0, and so 1 +θc= 1.We use the identity:
M= tanh(M+b) = (tanhM+ tanhb)(1 + tanhMtanhb), which leads to
M
M−M3
3 +b−b3 3
(1 + tanhMtanhb).
Cancel the factor ofM on both sides and rearrange, to obtain:
b∼ M3 3 +b3
3 −
M−M3
3 +b−b3
3 M b−M b3
3 −M b3 3 +. . .
.
Thereforeb∼M3/bfor smallb, M and by comparison with the defining relation, δ= 3.
If we set b ∼ M3 on the right hand side, we can verify all terms of order higher than O(M3) are neglected.
53
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72
Part III
The arrow of time
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Study notes for Statistical Physics:
A concise, unified overview of the subject
73
Classical treatment of the Hamiltonian N-body assembly
Chapter 6
Classical treatment of the Hamiltonian N -body assembly
In this section we discuss the behaviour of the assembly as a function of time. To do this, we formulate the microscopic description of an assembly in a way that is completely rigorous and fundamental yet which leads to some surprising results which do not appear to accord with everyday experience. Although we should note that our theory here is fundamental only insofar as that property is compatible with a classical description, we should emphasise two points. First, we shall as usual maintain contact with the quantum description, which should ensure that we do not do anything which is actually wrong. Second, the paradoxes which will arise do not depend on a quantum description for their resolution.
We may foreshadow the later paradoxical behaviour of the theoretical predictions by first discussing a simple, qualitative version of the reversibility paradox. Let us consider a box with an internal partition which divides it into two equal volumes, one of which contains a gas at (say) STP, and the other which is empty. The situation is illustrated in Fig. 6.1. Let us now imagine that the partition is broken in such a way that the gas can escape to the empty half of the box. Obviously this is just what will happen and the process will stop when the amount of gas in each half of the box is the same.
Yet when we try to describe this process at the macroscopic level, we run into a difficulty. The motion of each particle is governed by Newton’s laws and these are reversible in time. If we know the state of any one particle at any timet0(say), then we know its past historyt < t0 and its future behaviourt > t0
for all time. This is the deterministic picture. We can equally well run the clock backwards and the description of the particle motion will still be valid. Thus on a microscopic level, there would appear to be no reason to predict that a system would evolve irreversibly from a non-equilibrium state to an equilibrium one. Indeed, as we shall see, at this level of description it may not be possible to even say what we mean by an equilibrium state.
In the classical description, by ‘state of a particle’ we mean its instantaneous position and velocity.
The quantum description is, in this context, more difficult to envisage, because we have to think of the individual particles as undergoing transitions from one quantum state to another. These quantum states are the relevant solutions of the Schroedinger equation and this equation (which is equivalent to a statement of conservation of energy) is, like Newton’s laws, reversible in time. Thus, irrespective of whether we adopt a classical or a quantum description, at the microscopic level it is not immediately obvious why a system will evolve in one direction rather than another. This is the fundamental problem of statistical physics:
what determines the direction of time’s arrow? We shall consider this aspect further as we develop the theory in this chapter.
6.1 Hamilton’s equations and phase space
The treatment of this chapter will be based on Hamilton’s equations. It is assumed that the reader has met both the Lagrangian and Hamiltonian formulations of classical mechanics and so only the briefest of introductions will be given here.
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