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Appendix A Introduction to Statistical Physics

A.1 Introduction

Statistical physics and quantum physics constitute the foundation of the modern physics. They provide methods to study properties of matter.

Methods of statistical physics allow us to study macroscopic properties of large systems using microscopic mechanisms and structures proposed by quantum mechanics. Thanks to a combination of quantum mechanics and statistical mechanics, we have seen since the second half of the 20-th century spectacular discoveries and progress in modern physics, in particular in the field of condensed matter, which have radically changed our way of life. For the fundamentals of statistical physics and its application to condensed matter, the reader is referred to the book Statistical Physics: Fundamentals and Application to Condensed Matter [88].

This Appendix aims at recalling elements of statistical physics which are used throughout this book.

Statistical physics for systems at equilibrium is based on one single postulate called "the fundamental postulate" introduced in the case of an isolated system at equilibrium. The complete properties of an isolated system are deduced from this postulate. Other systems,

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ISBN 978-981-4877-42-8 (Hardcover), 978-1-003-12110-7 (eBook) www.jennystanford.com not isolated but in some special conditions, can be studied from methods derived from the fundamental postulate. One class of such systems includes systems in contact with a very large heat reservoir: a system of this class has a constant temperature fixed by the heat reservoir. The method used to study this class of systems is called the “canonical description” A second class of systems includes systems in contact with a large reservoir of heat and particles. A system of this class has a constant temperature and a constant chemical potential given by the reservoir. The number of particles of the system is not constant. This number fluctuates around a mean value when the system is at equilibrium. It is an "open system." The method used to study this class of systems is called the "grand- canonical description."

We consider a system of particles. In statistical physics the most fundamental quantity is the statistical entropy defined by

where P/ is the probability of the microscopic state / of the system and kB the Boltzmann constant. We shall use the statistical entropy to express various physical quantities in the following.

A.2 Isolated Systems: Microcanonical Description

A.2.1 Fundamental Postulate

A system is said "isolated” when it has no interaction with the remaining universe. It is obvious that such a definition is not rigorous: we should understand that interactions are so small that they are not observable and the parameters imposed on the system from the outside world such as energy E, volume V and number of particles N, are constant for all time.

The accessible microscopic states of an isolated system are the states which obey the external constraints. Let £2 be the total number of accessible microscopic states. The fundamental postulate of the statistical mechanics of systems at equilibrium states that

"All accessible microscopic states of an isolated system at equilibrium have the same probability."

According the above postulate, we have

for any accessible microscopic state /. The above probability is called "microcanonical probability.” Microscopic states which verify this probability constitute a "microcanonical ensemble." The description of properties of a system using the above probability is called "microcanonical description."

Statistical entropy S of an isolated system is thus

Equation (A.3) is called "microcanonical entropy."

The microcanonical temperature T is defined by

The microcanonical pressure p is defined by

The microcanonical chemical potential ц is defined by

We can show that these definitions correspond to physical quantities of the same names in thermodynamics. We can also show that the spontaneous evolution toward equilibrium of an isolated system when an external constraint is removed is always accompanied by an increase of statistical entropy S. Equilibrium is reached when S is maximum (see demonstration in Ref. [88]).

A.2.2 Applications

A.2.2.1 Two-level systems

We consider an isolated system of N independent, discernible particles. We suppose that N ^> 1. Each particle has two energy levels 2. The total energy of the system is equal to E. Using the microcanonical description, we calculate the total number of accessible microscopic states £2(£), the microcanonical entropy S(E) and the microcanonical temperature T as follows:

• The total number of accessible microscopic states £2(£):

System energy: E = Ahei + A/2e2 - constant, Ah=number of particles on eb Af2=number of particles on e2, we have N = N1 + N2 = constant; hence, E = Ahe 1 + (N — Ni)e2. As E is constant, N1 is thus determined. The total number of accessible microscopic states £2(£) is equal to the number of ways to choose N1 particles among N for the level ei. Thus, £2 = , =

JVj!(W-Wi)!'

• The microcanonical entropy S[E):

S = kB In Q = fcB[ln N - In Ah! — [N - Ah)!. Using the Stirling formula for N » 1: In N ~ N In N — N, we have S ~ kB[N In AZAT-Ah In Ni + Ni~[N — Ni) [N — Ni) + (N — N1)] = kB[N In N- N1 In /7Z/vf — N ln(AZ — Ah)]

• The microcanonical temperature T as a function of and e2:

T-1 = ^ = 7+rM- lnN' - 1+4N-

N1) + 1] = ^ln^; hence, ^ = 1п^огД = eXP(-^(el - e2)-

We obtain the relation of Ah and N2 as functions of T, ei and e2:

U. - iu exP(—^1—_ w exP(2fa)

JV1 - ,v l+exp(—^(«!—62) ~~ JV l+exp(2/)f)

• With e(> 0) = —ei = e2, we have

At low T, N1 -[1]■ N; hence, N2 —>■ 0. This result is obvious since particles occupy low-energy level at Г = 0. At high T, Ah —> A//2; hence, N2 —>■ N/2: particles are equally distributed on the two levels.

A.2.2.2 Classical ideal gas

We consider a classical ideal gas of N particles, of volume V. The gas is isolated with a total energy E. Using the microcanonical description, we calculate momenta). The number of states of energy < E is the number of states in the sphere of radius p = J2mE

where p, is the momentum of the z'-th particle. The integration on p, gives the volume of the sphere in 3N dimensions which is proportional to the radius to the power of 3 N, namely VN (2mE)3N/2 where VN is the integration over N positions. The number of states of energy equal to E, namely £2(E), is the number of states lying on the surface of the sphere of radius VN(2mE)3N/2, we have £2(£) a y«£3/v/2-i ^ yNg3N/2 (because N » 1). The microcanonical entropy is thus S = A In VN E3N/2 where A is a constant,

• The microcanonical temperature T:

We have T~x = = квЩ-, hence, E = NkBT, namely the

result of classical thermodynamics.

• The microcanonical pressure p:

We have £ = n = kBN/V; hence, pV = NkBT. This is the equation of state found in thermodynamics using the kinetic theory of gas.

A.3 Systems at Constant Temperature: Canonical Description

When a system is in contact with a heat reservoir much larger than the system, the reservoir imposes on the system its temperature T. The system energy is no more constant, it fluctuates by heat exchange with the reservoir. The equilibrium is reached when the system temperature is equal to that of the reservoir. If we know T, we can calculate the main properties of the system as seen in the following.

The probability of the microscopic state / in the canonical situation is given by

where /3 = and

We call Z the "partition function.” This function depends on external variables imposed on the system such as T, V (system volume) and N (number of particles).

The probability (A.8) is called "canonical probability.” The ensemble of microscopic states obeying this probability is called "canonical ensemble.” The description of properties of the system using (A.8) is called "canonical description.” We see below that we can express various physical quantities in terms of Z:

Average energy and heat capacity:

The average energy E of the system is

The heat capacity is

Canonical entropy:

Replacing Pi by (A.8) in (A.l), we have

Free energy:

The free energy F is defined by

As Z, F is a function of T, V and N. This definition allows us to write

or more often,

Canonical pressure p: p is defined by

Canonical chemical potential . ц is defined by

We can show that a system in a canonical situation tends to equilibrium, when an external constraint is removed, in the sense of decreasing F during the spontaneous evolution. The system reaches equilibrium when F is minimum [88].

A.3.1.1 Two-level systems

We consider the two-level system in A.2.2.1 using the canonical description at T. We calculate the following:

• The partition function Z:

We have Z = zN where z is the partition function of one particle, z = exp(—= exp(-^ei) + exp(-^e2) = 2 cosh(^e).

• The average energy E and the heat capacity Cy:

Ё = = -Nt tanh(/Je).

Г _ dE _ N c2 — dT — Wr7 cosh2(fit)

• The number of particles in each level:

We haveT = + N2e2 = e[-Ni + N2) = е(-Л/х + N-Л^) =

e(—2Ni + N). Using E given above, we have — Ne tanh(/Je) = e(N - 2Ni); hence, Ni = у [1 + tanh(^e)].

• At low T, N -[2]■ N (because tanh(^e) ->• 1); hence, N2 -[2] 0. At high T, Ni —[2]■ N/2 (because tanh(/Je) -[2] 0); hence, N2 -[2]■ N/2. We have the same results as by the microcanonical description.

A.3.1.2 Classical ideal gas

We calculate with the canonical description the partition function, the average energy and the pressure as follows:

We recover here the same result obtained by the microcanonical description given in the previous section.

  • [1] The microcanonical entropy: The classical phase space (see Section A.6 below) has 6А/dimensions (3N for particle positions and 3N for particle
  • [2] The partition function Z = zn/N (undiscernible particles): using du exp(-uu2) = n/a. • J = - = ±N; hence, J = • p = kBT^- = NkBT/V.
  • [3] The partition function Z = zn/N (undiscernible particles): using du exp(-uu2) = n/a. • J = - = ±N; hence, J = • p = kBT^- = NkBT/V.
  • [4] The partition function Z = zn/N (undiscernible particles): using du exp(-uu2) = n/a. • J = - = ±N; hence, J = • p = kBT^- = NkBT/V.
  • [5] The partition function Z = zn/N (undiscernible particles): using du exp(-uu2) = n/a. • J = - = ±N; hence, J = • p = kBT^- = NkBT/V.
  • [6] The partition function Z = zn/N (undiscernible particles): using du exp(-uu2) = n/a. • J = - = ±N; hence, J = • p = kBT^- = NkBT/V.
 
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