A Galilean covariant approach to classical mechanics of a single particle is described. Within the proposed formalism, all non-covariant force laws defining acting forces which become to be defined covariantly by some differential equations are rejected. Such an approach leads out of the standard classical mechanics and gives an example of non-Newtonian mechanics. It is shown that the exactly solvable linear system of differential equations defining forces contains the Galilean covariant description of harmonic oscillator as its particular case. Additionally, it is demonstrated that in Galilean covariant classical mechanics the validity of the second Newton law of dynamics implies the Hooke law and vice versa. It is shown that the kinetic and total energies transform differently with respect to the Galilean transformations

Horzela, Andrzej

Kapuscik, Edward

English

NASA Technical Reports Server

N93-27342
DOUBLE SIMPLE-HARMONIC-OSCILLATOR FORMULATION
OF THE THERMAL EQUILIBRIUM OF A FLUID
INTERACTING WITH A COHERENT SOURCE OF PHONONS
B. DeFacio and Alan Van Nevel
Department of Physics and Astronomy
Missouri University
Columbia, Missouri 65211, USA
and
O. Brander
Institute for Theoretical Physics
Chalmers University of Technology
8-41296 Gothenburg, Sweden
ABSTRACT
A formulation is given for a collection of phonons (sound) in a fluid at a non-zero
temperature which uses the simple harmonic oscillator twice; one to give a stochastic
thermal "noise" process and the other which generates a coherent Glauber state of phonons.
Simple thermodynamic observables axe calculated and the acoustic two point function,
"contrast" is presented. The role of "coherence" in an equilibrium system is clarified by
these results and the simple harmonic oscillator is a key structure in both the formulation
and the calculations.
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1. Introduction
The problem of understanding the thermal properties of a radiation field in a finite
volume is both old and subtle, l-a Here a sound wave propagating in a water will be studied
and the key issue will be the interaction of the sound radiation with the fluid matter and
with the wails of the container. The time scales of sound waves, v = 20 - 2 x 109Hz,
and those of the water molecules lead to adiabatic (isentropic, if approximately reversible)
thermodynamic processes rather than constant temperature transitions. 4 This work is a
special case of a project by one of us (AVN) which addresses the full nonlinear problem of
bubble formation by sound waves. The linearized problem will be studied here, where the
fluid has a coherent interaction with the sound radiation and an incoherent, or stochastic,
interaction with the reservoir.
The harmonic oscillator has played a central role in the coherent states 6-11 and will
be used for the coherent (Poisson) process describing the phonon radiation the sound field.
Since the reservoir is incoherent the total interaction with the fluid is partially coherent J 4'a 5
The reservoir is analogous to Feynman's rest of the universe 12 and Han, Kim and Noz _3
have shown the relation of this idea to quantum squeezed states and time-uncertainty.
The model presented here will use the simple-harmonic-oscillator twice: first to gener-
ate "stochastic or chaotic" noise and second to generate a Glauber coherent state of scalar,
longitudinal phonons. This is a more realistic model of noise, in that it has both coherent
2 ..... "
and a random components. It will be called partially coherent following a standard useage
in quantum optics. _i'he density, entropy free energy and a two-point function which gives
the acoustic contrast are all calculated. The reason that the SHO is so useful is that since
their Gaussian functions are dense in L 2, quantum mechanics guarantees that the crucial
interaction between the fluid and the sound radiation can be approximated by an infinite
collection of oscillators. This is why the approach of Planck 2 was correct even though
quantum mechanics was not yet created. Also, finite energy classical solutions will lie in
L 2 or at least in the Soboler space H 1 = L 2'1, which is the space where the "function"
310
and its "gradient" are square integrable.
In Sec. 2 the model will be presented, and the density p, the entropy S, the free
energy F and the pair correlation function g(2) are calculated for both single and N-mode
partially coherent states. In See. 3 the Conclusions and Outlook are presented.
2. The Model
In Fig. i, a schematic is given which shows a source of sound So (treated as a coherent
state of phonons) a fluid F in thermal contact with the reservoir R, which is much larger
than So or F. In general, phonons can enter the fluid from So and the fluid and reservoir
can exchange particles as well as heat but all other exchanges are negligible.
b
Source of Fluid, .i Reservoir, _ p
Phonons, F R
So
Fig. 1. Schematic of the system modeled. The source of a coherent state of phonons
is So, F is the stationary fluid volume and R is the reservoir which is much larger than the
sum of So and F. The wavy lines indicate boundaries which allow particles and energy to
pass.
311
The idea is a modification of one due to Kaup 16,1r that cavitons (here bubbles) are
solitons (here solitary waves). In other cases, Williamsson and Wieland is, Glimm 19, and
others have shown that many physically interesting model solutions for plasmas and clas-
sical fluids are nonlinear, coherent exitations of the medium. The formulation given can
easily be generalized to M1 independent random components of noise and Mz indepen-
dent, coherent components. Thus, solitons and some other nonlinear modes could easily
be added to the analysis.
The phonons are bosons so that their creation and destruction operators a*, a satisfy
the canonical commutation relations,
[a,a]= 0 = [a*,a'l, [a,a'] = 1 (1)
and a unique, translationally invariant Fock vacuum ]0 > exist s.t.
---4
a( x ,t){0>= o (2)
The astersik power of an operator is its adjoint (a* and a are not self-adjoint) and on a
complex number is its complex conjugate. Physically, the Fock vacuum is a quantum state
with no phonons. The number operator N is defined as
N = a*a (3)
and a number or Fock states is given by
a*) n
In >= n! 10 >
for each neZ+, the positive integers including zero. They are eigenfunctions of the number
operator with eigenvalues nez+. The Fock representation _'_F of the quantum Hilbert space
7-( is the L 2 closure of the linear span of the In > states. The inner-product of the Hilbert
space will be written as < .,- > and the inner-product compatible norm is written as
312
i
I1 I1= [< , >1-g, For any complex valued zeC' the unitary displacement operator, U(z),
acting on the Fock vacuum yields the minimum uncertainty coherent state [z > given by
Iz>: V(z)10>= e-N'/2Z z-
,=0 "_n! In > (4)
The Fock vacuum is the ground state of the SHO for the minimum uncertainty coherent
states which will be used here.
complex number z is written as
In terms of c-number coSrdinate q and momentum p the
z=(q,p)=q+ip
so that C 1 corresponds to the phase-space of the (1 - d) system.
coherent states are that
alz >-- zlz >
and
Zl,Z2 >_--- e-'ff(zl-z2)(zl-z2)e-_(zlz2-zlz2)
(5)
Two properties of the
(6)
(7)
From eq. (7) it is clear that the coherent are continuous in the label z and therefore are
an overcomplete family of states, OFS. The L2-closure of the linear span of the coherent
states Iz > provides a continuous representation of the physical Hilbert space which will
be written as 7-/c_.
A density operator is a positive, self-adjoint operator which satisfies
#=p=p* (s)
The expected value of an observable A = A* in a state ¢e7-/with corresponding density
operator p,# can be expressed as
< A >4=< ¢,A¢ >= Tr(p_A)
And additive thermal noise can be added "by hand."
given by
S = -kBTr[p lnp]
(9)
The entropy of the system, S, is
(10)
313
where kB is the Boltzman constant. In information theory, one can set kB = 1 In2 and still
use eq. (10). The entropy is obtained from maximizing eq. (10) subject to the constraints
Tr(p) = 1 (11)
and
Tr(pN) = c (12)
where ceR 1 is a parameter which labels the strength of the thermal state. If the thermal
noise is Gaussian, its density operator can be expressed as
p(c)= --Trcl/d_ze_l_12/Clz ><=1 (13)
In "HF this can be re-written as
_ z2np(c) = l,_c d2zc-I"i'/%-Izl' _ --_-.,1'_>< nl (14)
n----0
Using Fubini's theorem to interchange the integral and the infinite sum and then expressing
z in plane polar coordinates gives
p(c) = (c + 1)(1 + 1/c)"+a
in the Fock representation. A similar calculation of the entropy gives
s = k_[(c + 1)t_(c + 1) - c z_(_)] (16)
Sudarshan 15 and probably byThis sort of caieulation was given by Glanber ]°, Wolfa41 ......... _.... -:_
others. The nth-order correlation function, g(")(X1,..., X,), with Glauber's normalization
convention is
g(")(X_,..., X,) := G(",")(X_,. . .X2,)
2n
I-I
k=l
(17)
i
i
!
i
314
where the G(i'J)'s are Green's functions or correlation functions. This is a quantum gen-
eralization of the classical coherence degree -_ of Born and Wolf. 2° The visibility v of a
two-slit interference pattern is related to _12 by
v = 1 -t-I_,_(_)1 (18)
which physically represents the extremes in intensity. For the two-point function, g(2)(.),
the G (nm)'s are chosen s.t
Tr[pN(N- 1)] (19)
G(2)(p) = Tr(pN)]2 - 1
This object is proportional to Glauber's g(2) in ref. (10) but is not equal to his function
because of different normalizations..
In the case of the thermal states,
>< hi)=
the Fock representation of g(2) for n :fl 0 is given by
Tr[In >< nlN(N-1)] n 2 - n 1
-1--- n2 l-------n " (20)
Tr([n >< nlN)] 2
A coherent state is very different from eq. (20) since
Tr[Iz >< zlN(N-1)]
[Tr(ln >< nIN)] 2
- I = 0 , (21)g  )(Iz >< zl)=
i.e. the Glauber state is perfectly coherent.
For the thermal states, not surprisingly, g(2) has the opposite behavior from eq. (21)
because
g(2)[pTh(C)]=Th
[_n( 1 t'----a---1_"+l]n >< nN(N- 1)]Tr "_)\l-bl/c]
[ i / n+l
n --1 (22)
315
Defining
eq. (22) becomes
1(1)f.(c):= (c+1) 1+1/c n+l
E f.(c)( n2 - n) - E f.(c)n
(2) . - 1 (23)gTh(c) = 2 -- '
which is perfectly incoherent. Now a partially coherent state has a density operator which
is given by
p(z;c)=U(z)pTh(c)U*(z)= --1 [ d2xe_l.12/Clz +x >< z +x I
7re J
A calculation similar to the one outlined between eqs. (14) and (15) gives
Tr[p(z;c)N] = Tr{ l f d2
= Iz[2 + c
(24)
[(z + z)"(z* + x*)'e_l:+.12121n >< nlN fmn ] }x e -Ixl2/c E,n n!m!
rn
(25)
It is straightfoward to calculate the entropy of the partially coherent state from
=
=
= S(c)
= ka[(c+ 1)In(c+ 1)-cIn(c)] , (26)
where eq, (16) was used in the last two equalities. The pair correlation function, g(2), of a
partially coherent state is given by
g(2)[P(Z;C)] = l- ,z,2 )2c_ c + Izl2 ' (27)
in the coherent state basis. Let m > n with m, neZ+ and form the ninth matrix element
of the partially coherent density operator as
< n,p(z;c)m > = e -I_1_/(1+c) 1 1
1 + c (1 + llc)(m+nl 2)
(n! a/2 z pm
• ([4C Jr- C) n--m C( Izl2 ) (2s)l+c)
316
where Ptm(r) is the polynomial given by
r_ rn!
P'm(r)= _ k!(Z+ k)!(m- k)!rk
k=O
The thermal or noise density can be found from the c --* 0 limit of the previous equation
and is given by the familiar expression
< _,p(z;0)m>= _-JzJ'z"(z*)r" (29)
By calculating another Gaussian integral, an overlap of two partially coherent states is
found to be
TF[p(z 1;e 1)p(Z 2;c 2)] =
Clearly as C 1 ---t. C2 = C, eq. (30) reduces to
(1 + c 1 -3t- c2) (30)
1
Tr[p(O;c)p(O;c)] = 1 + 2"-------_ (31)
Remark: The non-zero part of the entropy S(c) = S(z; c) and the non-unit part of
Tr[(p(z,c)) 2] give ameasure of the nagnitude of the departure from apure coherent state.
The generalization to M modes, with M a positive, finite integer is straightforward. Let
(a, z) be M x 1 matrices, let (a*, z*) be 1 x M matrices, 1 the M x M unit matrix where
k = 1, 2,..., M labels which mode. Let B be a given non-singular, Hermitian, M x M,
positive, covariance matrix and express the M-mode Fock state as
M
I_,,... ,-m >= II (a*)-kl0>
k=l _
(32)
where nk is the occupation number of the kth-mode. The M-mode thermal density operator
is given by
PTh(13)=
e-a* ln( l +B-1)a
det(1 + 13) (33)
det(13) d2x'e-_"8-"xlx >< _[ ' (34)
317
where x = (zl,... ,aM) is the M-component, complex, coherent state amplitude. The
entropy of an M-mode phonon packet is given by
S(B) = kBTr[(t + B)In(1 + B) - 131nB] = S(x;B) (35)
for both the incoherent and the partially coherent cases. The pair correlation function g(2)
for an M-mode partially coherent system is
Tr[I3. B + 2z* -Bz]
¢_)(z;B) = (Tr(_) + Iz12)_ (3c)
which has the limits
g(2)(O,B) = 1 (37a)
and
¢2)(z,0) = 0 (37b)
To find the partition function for the partially coherent system one needs the mode energy
ek for the kth-mode which is real
e k = e k
and the M-mode vector
(38)
--_ e2
e = . (39)
eM
which is a vector-valued, intensive variable. The mode energies for the k th mode is
Ek = nkek = E_ (40)
where nk is the occupation number for the k th mode and
g= (EI,...,Em) (41)
is a real, vector-valued, extensive variable dual to eq. (39). In this more general case the
density operator p(z; 13) which maximizes the entropy
7 :
S = -kBTr[p Inp]
318
=
}
is subject to the constraints
and
T_(p) = I ,
Tr(pa) = z ,
Tr(pa*) : z*
(42@
(42b)
(42c)
Tr(pa" 7 a)= z* _e z+ e (42d)
Let _ be an m-vector with the value fl = 1/kBT for each component and express the
given covariance matrix for bosons as
--4---4
_J = _J(fl) = (e- fi' t: _ 1)-1 = (e-fiE _ 1)-' (43)
( A similar argument for Fermi-Dirac particles would replace -1 by +1 in eq.
this is not needed here.) The partition function Z(fl, V) is now
(1)Z(fl, V) = det 1- e -fiE
(43) but
(44)
at thermal equilibrium and V is the volume of the fluid plus radiation. From eq. (44) all
of the equilibrium thermodynamic quantities can be calculated, for example the average
energy g is
when classically
__, Ere -fl E_
g = _ _ ran(z) (45a)
z Off
Z(fi, V) = E e-fiE" (45b)
T
and in quantum statistical mechanics a Trace over matrix elements of e -fill is taken. The
pressure p is
10ln(Z) (45c)
P-_ ov
319
and the (reversible) differential work dW is
dW = -pdV (45d)
and the previous equation can be used to obtain
dW = - l Oln( Z) dv
fl OV
The entropy of eq. (10) can also be written as
s = kB[t,,(Z)+ Ze-] (45e)
and the Helmholtz free energy is
F(/_, V) = -_In[Z(_, V)] (45f)
The coherent state density operator can be given as
p(z) = det [(1- e-[tE)e -(a'-z')_E(a-z)] (46)
Near equilibrium, a Kubo linear response theory 21 can be established by studying small
deviations from equilibrium where the fluctuations will be equal to the dissipations. This
exercise will be left to a future project.
3. Conclusions and Outlook
A two-component thermodynamics was formulated for a fluid in thermal equilibrium
with a reservoir radiated by a coherent state of phonons. One future project will be to
derive the Kubo fluctuation-dissipation theorem for this system, another will be to compare
and contrast these results with both the Langevin equation and the stochastic quantization
approaches. These future studies should illuminate (or simplify) the lattice Monte Carlo
methods.
Much remains to learned from harmonic, SHO, systems.
320
4. Acknowledgements
This work was supported in part by AFOSR grants 90-307 and 91-203, the Department
of Physics and Astronomy at Missouri University and the Institute for Theoretical Physics
at Chalmers University of Technology. Professor Karl Erick Erickson of Chalmers suggested
part of this analysis to one of us (BDF) long ago in 1983-4. He is thanked for sharing his
wisdom. Brander and DeFacio (unpublished) have also applied this analysis to polarized
electromagnetic scattering.
321
References
1. G. Kirchhoff, Ann Phys. Chem 109, 275 (1860).
2. M. Planck, Verh. Deutsch Phys. Ges 2, 202 (1900) and ibid 2, 237 (1900).
3. A. Einstein, Ann. Phys. (Leipzig) 22,569 (1907).
4. G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press,
Cambridge and New York, 1967).
5. Alan Van Nevel, Bubble formation in water under sonification as a nonlinear, coherent
process (MS Thesis, Missouri University, Columbia, Missouri, in progress).
6. E. SchrSdinger, Naturwissenschaften 14, 664 (1926).
7. J. Von Neumann, Mathematical Foundations of Quantum Mechanics (Springer, Berlin,
1932).
8. J.R. Klauder, Ann. Phys. (NY) 11, 123 (1960).
9. V. Bargmann, Commun. Pure and Appl. Math. 14, 187 (1961).
10. R.J. Glauber, Phys. Rev. 130, 2529 (1963) and ibid 131, 2766 (1963).
11. J.R. Klauder and B.-S. Skagerstam, Coherent States World. Scientific (Singapore,
1985): contains a 115 page introduction and 69 reprints.
12. R.P. Feynman, Statistical Mechanics (Benjamin/Cummings, Reading, MA, 1972).
13. D. Han, Y.S. Kim and M. Noz, NASA CP 3135 269 (1992).
14. E.L. Wolf, J. Opt Soc. Am. 68, 6 (1978); ibid A3, 76 (1986).
15. E.G.C. Sudarshan, Int. J. Theor. Phys. 20, 186 (1981).
16. D.J. Kaup, Phys. Fluids 31, 1645 (1988).
17. D.J. Kaup, Phys. Rev. Lett. 59, 2063 (1987).
18. H. Williamson and J. Widand, Coherent Nonlinear Interactions of Waves in Plasmas
(Pergammon, New York, 1977).
19. J. Glimm, SIAM Review 33, 626 (1991).
20. M. Born and E.L. Wolf, Principles of Optics 4E (Pergammon, New York, 1970).
21. R. Kubo, J. Phys. Soc. Japan 12, 570 (1957).
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Galilean covariant harmonic oscillator

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