Lecture 3: Fluctuations, Microcanonical Ensemble

Chapter I. Basic Principles of Stat Mechanics

A.G. Petukhov, PHYS 743

August 30, 2017

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 1 / 12 Probability

Consider a system described by a random f variable which can assume M discrete values (e.g. a tossed coin - 2 values, a rolled die - 6 values etc.). A set of N experiments on such a system (e.g. tossing a coin N times, rolling a die N times etc. ) is an example of a statistical ensemble. The possible outcomes of a single experiment are f1, f2, ..., fM . Let us suppose that the number of experiments giving a particular outcome xm is Nm (numberofoccurrences or frequency) so that

M X Nm = N (1) m=1 The probability of each outcome is deﬁned as

Nm wm = lim (2) N→∞ N

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 2 / 12 Expectation Value It follows from Eq. (1) that wm satisfies the normalization condition: X wm = 1 m For a finite N one can define the statistical average or expectation value:

1 X hfi = N f N m m m This deﬁnition holds for N → ∞. Thus X hfi = wnfn n We can also calculate the expectation value of any function F (f) of the random variable f: X hF i = wnF (fn) (3) n

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 3 / 12 Let us consider two separated subsystems of a large system. We assume that the interaction between the particles is short-range. It means that the two subsystems do not interact with each other and only weakly interact with the environment. Indeed, the number of particles (interacting with the environment) at the surface is much smaller than in the bulk and the two systems can be considered as quasi-closed. The distribution functions are deﬁned in diﬀerent domains (q1, p1) and (q2, p2) and the probability:

dw = dw1dw2 = ρ1(q1, p1) · ρ2(q2, p2)dq1dp1 · dq2dp2

Distribution Function of Two Weakly Interacting Systems Eqs. (2) and (3) can be straightforwardly generalized to the cases of many random variables and continuous distributions: Z hF i = F (x1, x2, . . . , xk)dw(x1, x2, . . . , xk) Z = F (x1, x2, . . . , xk)ρ(x1, x2, . . . , xk)dx1, dx2, . . . , dxk,

where ρ(x) is the familiar distribution function.

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 4 / 12 The distribution functions are deﬁned in diﬀerent domains (q1, p1) and (q2, p2) and the probability:

dw = dw1dw2 = ρ1(q1, p1) · ρ2(q2, p2)dq1dp1 · dq2dp2

where ρ(x) is the familiar distribution function. Let us consider two separated subsystems of a large system. We assume that the interaction between the particles is short-range. It means that the two subsystems do not interact with each other and only weakly interact with the environment. Indeed, the number of particles (interacting with the environment) at the surface is much smaller than in the bulk and the two systems can be considered as quasi-closed.

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 4 / 12 Distribution Function of Two Weakly Interacting Systems Eqs. (2) and (3) can be straightforwardly generalized to the cases of many random variables and continuous distributions: Z hF i = F (x1, x2, . . . , xk)dw(x1, x2, . . . , xk) Z = F (x1, x2, . . . , xk)ρ(x1, x2, . . . , xk)dx1, dx2, . . . , dxk,

dw = dw1dw2 = ρ1(q1, p1) · ρ2(q2, p2)dq1dp1 · dq2dp2

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 4 / 12 Statistical Independence

Thus

ρ12 = ρ1 · ρ2 ⇔ probability of independent events is a product

Expectation value: Z hf1 · f2i = f1(q1, p1) · f2(q2, p2)dq1dp1 · dq2dp2 = hf1i · hf2i (4)

Introducing average deviation:

h∆fi = hf − hfii = hfi − hfi = 0 (5)

By virtue of Eqs (4) and (5)

h∆f1 · ∆f2i = 0 (6)

Systems satisfying Eq. (6) are called statistically independent

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 5 / 12 Fluctuations

To describe ﬂuctuations let us deﬁne mean-squared deviation ≡ variance:

(∆f)2 = (f − hfi)2 = f 2 − 2 hfi hfi + hfi2 = f 2 − hfi2 ,

root-mean-squared deviation δf = ph(∆f)2i, and relative ﬂuctuation δf/ hfi.

Let f be an additive (extensive) quantity such as f1+2 = f1 + f2, e.g energy, mass, number of particles, entropy (see below) etc. Let us divide our system into a large number N → ∞ of almost identical subsystems. Then N X hfi = hfii ' N hfii i=1

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 6 / 12 Fluctuations, cont’d Let us calculate the variance of f:

* N N !2+ * N !2+ 2 X X X (∆f) = fi − hfii = ∆fi i=1 i=1 i=1 N N X D 2E X = (∆fi) + h∆fi · ∆fki i=1 i,k(i6=k) N X D 2E D 2E = (∆fi) 'N (∆fi) i=1 Therefore the relative ﬂuctuation: √ δf N δf 1 = i ' √ hfi N hfii N N can be considered as a number of particles. That is why the ﬂuctuations of macroscopic quantities are very small

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 7 / 12 Signiﬁcance of Energy

As we know for any mechanical system there are 2s − 1 integrals of dρ motion. To satisfy the Liouville’s theorem ( dt = 0) the distribution function must depend on p(t) and q(t) through integrals of motion. Since ρ1+2 = ρ1 · ρ2 then log ρ12 = log ρ1 + log ρ2, i.e. logarithm of the distribution function must be additive. As such the distribution function must depend only on additive integrals of motion, i.e. E, P , and M. For the system at rest ρ depends only on the energy E.

Consider a subsystem a with energy Ea of the large system A. The only possible additive function of energy for log ρa is

log ρa = αa − βEa

The constant αa can be found from the normalization condition and the distribution ρa is called the Gibbs distribution, which we will study in details later.

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 8 / 12 Statistical Postulate

(i) (i) (i) (i) Consider a point (q1 , q2 , . . . p1 , p2 ... ) in the phase space. In a discretized phase space a cell of the volume hs surrounding this point is called a microstate. In quantum mechanics a micro state is naturally deﬁned as an eigentstate |ni of the Hamiltonian corresponding to energy En

Postulate of equal a priori probability: each accessible microstate within the energy interval ∆E occurs with equal probability 1/∆Γ(E) where ∆Γ(E) represents the total number of such states. The quantity ∆Γ is called multiplicity or statistical weight.

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 9 / 12 Microcanonical Distribution

Consider a closed system with phase trajectory belonging to a hyper-surface H(p, q) = E. According to the statistical postulate:

const, q, p ∈ hypersurface H(p, q) = E ρ = 0, otherwise

To ensure that Z ρdpdq/hs = 1, (7)

i.e. ﬁnite, ρ must be expressed through δ-function:

ρ(p, q) = Cδ [E − H(p, q)] (8)

Eq. (8) describes microcanonical distribution. The constant C can be found from the normalization condition (7).

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 10 / 12 The number of microstates in the phase volume enclosed by this surface reads: Z Z Γ(E) = ··· Θ[E−H(p, q)]dq·dp/hs, (9)

where 1, x > 0 Θ(x) = 0, x < 0 is the Heaviside step function. Diﬀerentiating Eq. (9) over E and using dΘ(x)/dx = δ(x) we obtain

dΓ(E) Z Z = ··· δ[E − H(p, q)]dq · dp/hs ≡ g(E) (10) dE

Microcanonical Distribution, cont’d Normalization constant C can be calculated as H( p,q) = E follows. Note, that hypersurface H(p, q) = E is closed for a ﬁnite system because q and p are bound.

Γ(E) E + ΔE

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 11 / 12 where 1, x > 0 Θ(x) = 0, x < 0 is the Heaviside step function. Diﬀerentiating Eq. (9) over E and using dΘ(x)/dx = δ(x) we obtain

dΓ(E) Z Z = ··· δ[E − H(p, q)]dq · dp/hs ≡ g(E) (10) dE

Microcanonical Distribution, cont’d Normalization constant C can be calculated as H( p,q) = E follows. Note, that hypersurface H(p, q) = E is closed for a ﬁnite system because q and p are bound. The number of microstates in the

Γ(E) E + ΔE phase volume enclosed by this surface reads: Z Z Γ(E) = ··· Θ[E−H(p, q)]dq·dp/hs, (9)

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 11 / 12 Microcanonical Distribution, cont’d Normalization constant C can be calculated as H( p,q) = E follows. Note, that hypersurface H(p, q) = E is closed for a ﬁnite system because q and p are bound. The number of microstates in the

Γ(E) E + ΔE phase volume enclosed by this surface reads: Z Z Γ(E) = ··· Θ[E−H(p, q)]dq·dp/hs, (9)

where 1, x > 0 Θ(x) = 0, x < 0 is the Heaviside step function. Diﬀerentiating Eq. (9) over E and using dΘ(x)/dx = δ(x) we obtain

dΓ(E) Z Z = ··· δ[E − H(p, q)]dq · dp/hs ≡ g(E) (10) dE

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 11 / 12 The multiplicity ∆Γ(E) is the total number of states in the volume between the surfaces E and E + ∆E. We can express it through the density of states as:

dΓ(E) ∆Γ(E) = ∆E = g(E)∆E dE

Microcanonical Distribution, cont’d

The quantity

dΓ(E) g(E) ≡ dE is called the density of states. Using Eqs. (7), (8) and (10) we ﬁnally obtain 1 C = g(E)

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 12 / 12 Microcanonical Distribution, cont’d

The quantity

dΓ(E) g(E) ≡ dE is called the density of states. Using Eqs. (7), (8) and (10) we ﬁnally obtain 1 C = g(E)

The multiplicity ∆Γ(E) is the total number of states in the volume between the surfaces E and E + ∆E. We can express it through the density of states as:

dΓ(E) ∆Γ(E) = ∆E = g(E)∆E dE

Chapter I. Basic Principles of Stat Mechanics A.G.Lecture Petukhov, 3: Fluctuations, PHYS 743 Microcanonical Ensemble August 30, 2017 12 / 12