3.16 † Equilibrium of an Accelerating Body

Sinceρcmandρcm^a are numerically equal, this may also be written as Ucm=−∂lnZ_cm^a

∂ β ^(3.253)

We note thatU_cmreflects the average behavior of 3(N−1)mechanical degrees of freedom. The macroscopic motion of the center of mass accounts for the remaining three degrees of freedom.

3.16 †Equilibrium of an Accelerating Body 163

a, x^a

y^a

y x ater W

Fig. 3.7 A rotating bucket of water as observed from the inertial frameO^aand the frameOthat rotates with the bucket.

the center of mass of the particles. This means that the constraint (3.243) is absent here. Instead of (3.242), therefore, we have

ρ r^N,p^N

= 1 Z

e^−βH

h^3NP and Z= 1 h^3NP

e^−βHdr^Ndp^N (3.256) for a system in thermal contact with the surroundings.

3.16.2 † Rotation

Let us consider the case of rotation without any linear translation. Once again, we adopt a frame of referenceO that is attached to the container rotating with respect to the inertial frameO^a. For simplicity, we assume that these two frames share the same origin. Figure3.7illustrates the situation.

The frame O clearly is not an inertial frame: A particle at rest in the rotating frame is accelerating with respect to the inertial frame. In general, the effect of acceleration is felt asapparent forces. In the case of rotation, they are thecentrifu- gal force, theCoriolis force, and theEuler forceas we shall see in Example3.8.

This last terminology is due to Ref. [2].

It is clear that an external fieldψthat is time independent inO^avaries with time inOin general. In this section, therefore, we assume that the effect ofψis negligible in comparison to the intermolecular forcesandthe apparent forces.

Again, we need to establish the relationship between the velocity vector of a particle as measured inO^aand that inO. For this purpose, suppose that the posi- tion vector of a particle, as measured inO, changed fromrtor+drduring a time duration dt. If we denote the same vectors as observed inO^abyr^aandr^a+dr^a, respectively, then,

r=r^a (3.257)

because bothrandr^arefer to the same arrow in space. But the final vectorr+dris not equal tor^a+dr^abecause of the rotation ofOwith respect toO^a. We stress that dr^ais the changeas observed inO^aof the vectorr^a. This change is brought about by two contributions, one is the change dras observed inOof the vectorrand the other is the rotation ofOwith respect toO^a. This is why dr^a=dr.

To make things easier, let us first suppose thatrremained constant inO. Denoting the rotation ofO with respect toO^abydφas we have done in Sect.1.8.2, we see that

dr^a=dφ×r, (3.258)

and hence

r^a+dr^a=r+dφ×r. (3.259)

Ifrdid not remain constant but becamer+dr, we have only to replacerbyr+dr and obtain

r^a+dr^a= (r+dr) +dφ×(r+dr). (3.260) Now we cancelr^aandrusing (3.257) and divide the resulting expression by dt to arrive at

v^a=v+Ω×r, (3.261)

where we ignored the higher order termdφ×dr. The new vectorΩ:=dφ/dt is called theangular velocityofOwith respect toO^a.

Using (3.257) and (3.261) in (3.222) withψ_≡0, we find L^a=

N

∑

i=1

1

2m_i||vi||²−φ r^N +

N i=1

∑

m_ivi·^Ω×ri+

N i=1

∑

1

2m_i||^Ω×ri||². (3.262)

Example 3.8. Apparent forces: LetN=1 in (3.262):

L^a=1

2m||v||²+mv·^Ω×r+1

2m||^Ω×r||², (3.263) from which we derive Lagrange’s equation of motion of the particle inO. This allows us to identify the expressions for the apparent forces.

From (3.263),

∂L^a

∂v =mv+mΩ×r. (3.264)

To compute∂^La/∂r, we make use of (A.32):

∂

∂r(v·^Ω×r) = ∂

∂r(r·v×^Ω) =v×^Ω. (3.265) We also note that

||^Ω×r||²=||^Ω||²||r||²sin²θ=||^Ω||²||r||²(1−cos²θ) =||^Ω||²||r||²−(Ω·r)². (3.266)

3.16 †Equilibrium of an Accelerating Body 165

Thus, 1 2

∂_||Ω×r||²

∂r =||^Ω||²r−(Ω·r)Ω=||^Ω||²(r−r) =||^Ω||²r_⊥, (3.267) where we defined

r:= ^Ω

||^Ω||||r||cosθ and r_⊥:=r−r. (3.268) Geometrically, this decomposesrinto a part (r) that is parallel toΩand the other (r_⊥) that is perpendicular toΩas seen from Fig.3.8.

From Lagrange’s equation of motion, we obtain

m˙v+m_Ω˙ ×r+mΩ×v=mv×^Ω+m||^Ω||²r_⊥. (3.269) Solving form˙v,

m˙v=2mv×^Ω+m||^Ω||²r_⊥−m_Ω˙ ×r. (3.270) In this equation, the terms on the right are the apparent forces. From the left, they are the Coriolis force, the centrifugal force, and the Euler force. Being perpendicular tov, the Coriolis force does not perform any work on the parti- cle.

Coming back to (3.262), we simply adopt it as the LagrangianL inO without further modifications. Thus,

p_i:=∂L

∂vi

=m_i(vi+Ω×r_i) =m_iv^a_i =p^a_i , (3.271) and hence

H:=

N i=1

∑

p_i·vi−L=

N

∑

i=1

p^a_i·(v^a_i−^Ω×ri)−L^a. (3.272)

.... ...

...

...

...

...

r

r r

rcos

Fig. 3.8 Decomposition ofrintor_⊥andr.

Since the first and the third terms taken together isH^a, it may be replaced by the expression given in (3.225), where we recall thatψ≡0 in this section. Noting once again thatr^a_i =riandp^a_i =p_i, we find

H=

N i=1

∑

||p_i||² 2mi

+φ(r^N)−

N i=1

∑

p_i·^Ω×r_i. (3.273) Using the vector identity (A.32) and the definition (1.145) of the total angular momentum, we have

N i=1

∑

p_i·^Ω×ri=Ω·

N i=1

∑

ri×p_i=Ω·M, (3.274) and hence

H=

N i=1

∑

||p_i||²

2m_i +φ(r^N)−^Ω·M. (3.275) WhenΩis independent of time, the rotation is said to be uniform. For auniform rotation,H has no explicit time dependence and the system is expected to reach equilibrium eventually. The statistical distribution is then given by

ρ(r^N,p^N) = 1 Z

e^−βH

h^3NP and Z= 1 h^3NP

e^−βHdr^Ndp^N. (3.276) Whilep_iis the generalized momentum conjugate tori, it isnotthe mechanical momentumm_ivian observer inO would compute based on the values ofv^N mea- sured in that frame. The situation is entirely analogous to (3.231). To extract from Han energy term this observer would compute, we use (3.271) to eliminatep_ifrom (3.275) in favor of themechanical momentumdefined byπi:=m_ivi. This leads to

E=

N i=1

∑

||^πi||² 2mi

+φ(r^N)−

N i=1

∑

1

2m_i||^Ω×ri||²=:E₀+ψcp, (3.277) where

E₀:=

N i=1

∑

||^πi||² 2mi

+φ(r^N) and ψcp:=−

N i=1

∑

1

2m_i||^Ω×ri||². (3.278) The expression (3.277) is numerically equal toH in (3.275) but it is not a Hamil- tonian since E is given in terms of the mechanical rather than the generalized momenta. Nevertheless, the expression is illuminating. In particular,E0is the energy an observer inOwould compute based on the values ofr^Nandπ^Nmeasured in this frame without any regard to the fact thatOis rotating. Equation (3.277) also makes it clear that the effect of rotation of our coordinate system manifests itself only through theapparent external fieldψcpcalled thecentrifugal potential energy.

3.17 Frequently Used Symbols 167

Exercise 3.23.Verify (3.277). ///

Because of its intuitive appeal, it will be desirable to use E rather than H to express the statistical distribution. To find the functional form of this distribution, let

ρ^′(r^N,π^N)dr^Ndπ^N (3.279) denote the probability of finding the system of interest within the infinitesimal vol- ume element dr^Ndπ^N taken around the point(r^N,π^N)in the space spanned by 2N vectorsr^N andπ^N. From (3.271), we see that the Jacobian of the coordinate trans- formation from(r^N,p^N)to(r^N,π^N)is unity. As in Sect.3.15.3, this implies thatρ^′ is simplyρ expressed in terms ofr^N andπ^N. From (3.276), we arrive at

ρ^′(r^N,π^N) = 1 Z^′

e^{−β(E0+ψcp)}

h^3NP and Z^′:= 1 h^3NP

e^{−β(E0+ψcp)}dr^Ndπ^N. (3.280)

3.17 Frequently Used Symbols

A, ensemble average of a dynamical variableA.

f , the number of mechanical degrees of freedom.

h, Planck constant. 6.626×10⁻³⁴(J·sec).

k_B, Boltzmann constant, 1.3806×10⁻²³J/K.

m_i, mass of theith particle.

p_i, generalized momentum conjugate toq_i. p^f , collective notation forp1, . . . ,p_f. p_i, linear momentum of theith particle.

p^N , collective notation forp₁, . . . ,p_N. q_i, theith generalized coordinate.

q^f , collective notation forq1, . . . ,q_f. ri, position vector of theith particle.

r^N , collective notation forr1, . . . ,rN. t, time.

vi, velocity vector of theith particle.

v^N , collective notation forv1, . . . ,vN. A, a generic dynamical variable.

C, normalization constant forρ. C_V , constant volume heat capacity.

F , Helmholtz free energy.

H, Hamiltonian.

L, Lagrangian.

M , total mass of a many-particle system.

N, total number of particles in a system.

S, entropy.

T , absolute temperature.

U , internal energy.

V , volume.

Z, canonical partition function.

N , total number of copies in a statistical ensemble.

P , the number of permutations of identical particles.

V , internal virial.

β , 1/kBT.

δ(x), Diracδ-function.

ρ , statistical weight.

π, mechanical momentum.

φ , potential energy due to interparticle interactions.

ψ , potential energy due to an external field.

Λ , thermal wavelengthh/√

2πmk_BT of a particle of massm.

Ω, angular velocity.

References and Further Reading

1. Gibbs J W (1981) Elementary principles in statistical mechanics. Ox Bow Press, Connecticut Our Sect.3.9followed Chap. 4 of the book, where the canonical ensemble was introduced. The same chapter demonstrates the relationship between statistical mechanics and thermodynamics and provides further motivation for the canonical ensemble.

2. Lanczos C (1986) The variational principles of mechanics. Dover, New York The phrase “Euler force” is introduced in p. 103.

3. Landau L D, Lifshitz E M (1980) Statistical physics: Part 1, 3rd edn. Pergamon Press, New York

In developing the basic principles of statistical mechanics, we loosely followed Chap. 1 of the book. Our treatment of a rotating body is based on their treatment of the subject, in particular, Sects. 26 and 34.

4. Schr¨odinger E (1989) Statistical thermodynamics. Dover, New York

Chapter 3 of the book provides a detailed discussion on the meaning of setting the constant term in (3.67) to zero. He uses Boltzmann’s entropy formula rather than that of Gibbs. But, the former can be derived from the latter as we shall see later.

5. Tolman R C (1979) The principles of statistical mechanics. Dover, New York

For an extended discussion on the statistical approach and its validity, see Chap. 3. A brief summary is in Sect. 25.

6. Whitaker S (1992) Introduction to fluid mechanics. Krieger Publishing Company, Florida It is far more natural and physically compelling to write down laws of physics, such as the conservation of mass, Newton’s equation of motion, and the first law of thermodynamics, for a (possibly moving and deforming) body than to do so for a control volume fixed in space.

A derivation of the equation of continuity from this more satisfactory stand point is found in Chap. 3.

Chapter 4