Instabilities in classical and quantum fluids

PHYSICD ELSEVIER

Physica D 119 (1998) 381-397

Instabilities in classical and quantum fluids A. Muriel a.., E Esguerra a, L. Jirkovsky a,b, M. Dresden ,,c a Center for Fluid Dynamics (World Laborato~), College, Laguna, UniversiO, of the Philippines at Los Bagos, Philippines b National Institute of Physics, University of the Philippines at Diliman, Philippines c Stanford Linear Accelerator, SLAC Bin 61, Stanford Universi~, PO Box 4349, Stanford, CA 94309, USA

Received l December 1997; accepted 27 January 1998 Communicated by A.M. Albano

Abstract A formal analysis of the time evolution of the one-particle momentum distribution is introduced to demonstrate that runaway solutions, violating the conservation of momentum, occur when non-central pair potentials are used in classical systems. It is suggested that the problem is resolvable by including internal degres of freedom. Similarly, when non-Hermitean Hamiltonians are introduced in quantum systems, runaway behavior is also found. These suspected instabilities are interpreted to indicate the onset of turbulence and point out the need for new approaches in describing the many-body behaviour of classical systems with non-central pair potentials, and the quantum description of systems with non-Hermitean Hamiltonians. © 1998 Elsevier Science B.V. Keywords: Turbulence; Instabilities

I. Introduction In a recent series of three papers [1-3] based on two older papers [4,5], projection techniques were used to arrive at some new conclusions on many-body physics. Foremost among the conclusions is the occurence of instabilities that may well be interpreted as the onset of turbulence. The physical reason for the onset of unstable behavior is traced to the excitation of the internal degrees of freedom of molecules participating in the motion of a fluid. There exists that threshold energy, the energy gap between the ground state and the excited state, that must be overcome before turbulence begins, according to this microscopic theory. This is a physical criterion for the development of "high energy hydrodynamics" which seems to explain the onset problem of turbulence [6,7]. The energy gap we speak of could be the difference between rotational energy states of a molecule, which is a fraction of the thermal energy k T (k is Boltzmann constant and T is the temperature) of a fluid. This energy gap may be overcome by thermal or mechanical perturbation. * Corresponding author. E-mail: [email protected]. * Deceased. 0167-2789198l$19.00 © 1998 Elsevier Science B.V. All rights reserved PH SO I 67-2789(98)0001 7-7

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A. Muriel et al./Physica D 119 (1998) 381-397

In a fluid in laminar flow, the internal energy states of the molecules are not excited by collisions or external fields. The ordinary continuum model is adequate, that which we have begun to call '° low energy hydrodynamics". In the continuum model, the Navier-Stokes equations says it all, including, it has been believed, the origin of turbulence. This view makes turbulence a mathematical flow problem independent of molecular structure. The reality of atoms and molecules is encapsulated in the macroscopic properties of matter, the transport coefficients, of which viscosity is the most important, as in the Reynolds number, p L y ~ o , where p, L, v, and r/are the density, a typical length, fluid velocity and the viscosity, respectively. Despite the success of conventional hydrodynamic theory, it is perhaps fair to say [8,9] that the onset of turbalence is not completely understood. In distinct opposition to the conventional continuum model, our microscopic view considers molecular structure on the matter of transition from laminar flow to turbulance. In turn, this means that we cannot ignore quantum mechanics, for quantum machanics determines the structure of molecules. It is therefore curious to observe that mathematical approaches reminiscent of quantum mechanics have already been in use in the theoretical studies of turbulence [10-12]. But these are works applied to the fundamental equation of hydrodynamics [13], the Navier-Stokes equations. Our own view is that such methods must indeed be very useful, especially when we apply them to the more fundamental equation, the many-body Schr6dinger equation. By so doing, we acknowledge the importance and relevance of quantum mechanics, most properly invoked when one considers the structure of matter. With such an approach,we pull the study of turbulence away from applied mathematics alone; we bring it back to physics, where we think it may properly belong. By using quantum mechanics, the application of quantum mechanical and field theoretical methods renders the subject of turbulence in harmony with physics. Our results so far [ 1-3] have been developed. In pieces in this paper we take a more abstract approach to consider two questions. First, we address the following problem: Is there a more formal way of showing that a system of bodies interacting with non-central pair forces in indeed markedly different? In [ 1], it was hinted that an approximate kinetic theory of N bodies with non-central pair potential will result in an instability. This instability is seen in the possibility of arriving at new kinetic equations with qualitatively different solutions corresponding to (a) steady flow, (b) periodic flow, (c) quasi-periodic flow, and (d) highly irregular, probably random, and possibly turbulent solutions, depending on the degree of asymmetry of the pair potential. In this work, we show that an exact formal solution for the single momentum distribution function shows runaway behavior when the pair-potential is non-central. We display in Section 3 the time evolution of the one-particle momentum distribution function and its runaway features. Having done so, we then explain that a naive straightforward treatment of a system with non-central potential results in runaway behavior that can only be corrected by admitting that once a non-central potential is assumed, we are obliged to consider the existence of internal degrees of freedom, from which a desired reformulation of the problem must start. One ends up with a new problem akin perhaps to a Liouvillian formulation of interacting soft, deformable tops, a most formidable problem when treated classically, but more tractable quantum mechanically. In developing our rationale, we have departed as far as we could from the continuum model. In fact, a molecular kinetic approach is already part of the literature in a context that is not considered related to turbulence. We refer to modifications of the Boltzmann collision terms to include the internal degrees of freedom of molecules [14-16]. In these studies, the quantum nature of the rotational states of molecules has been included. But such semi-classical approaches have not been previously invoked to be related to instabilities, much less turbulence. Furthermore, we do not use a Boltzmann approach. Second, we found in [2] that a non-Hermitean Hamiltonian model produces unstable behavior in a simple quantum system, like our two-level atom in a mean field approximation. A non-Hermitean Hamiltonian is usually a representation of decay of states, classically interpreted as a dissipative system. Turbulence occurs in driven, dissipative systems. Hence, our quantum model may be relevant to the demonstration of the occurrence of instability that may signify the onset of turbulence. Is it possible to make this feature of non-Hermiticity a more general result?

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We show this in Section (4) with the conclusion that an anti-Hermitean Hamiltonian produces unstable behavior. We now explain our approach in addressing the two problems mentioned above. The total momentum or current of an isolated and closed system of N bodies is either zero or a constant, for both classical and quantum systems. In the classical case, the total momentum is given by

,,) while for the quantum case, we have (J(t)) ---- ~

'f

d r ( ~ ( r , t)*VqJ(r, t) - ~O(r, t ) v q / ( r , t)*),

(2)

where f ( r , p, t), ~o(p, t), and qJ (r, t) are the one-particle distribution function, the momentum distribution function, and the one-particle reduced wave function, respectively. Given that the above averages are either zero or a constant, why should there be an interest in the above quantities? In [3], we showed that the classical local current J(r, t), a hydrodynamic variable, is differentiable for all but two cases: when the central pair potential becomes infinite, as in a hard sphere potential, or when the initial data, such as the spatial probability, contain discontinuities. In addition, using a well-defined criterion [3,17], it is suspected that whenever the current becomes infinite, or when it is no longer differentiable, the system is probably turbulent, or in any case unstable. We have earlier found two cases of instability: (1) when the pair potential is non-central [ 1], and (2) when the many-body collisions are great enough to excite internal degrees of freedom such as rotations, or spins [2]. In this paper, we will concentrate on the analytic properties of the current. If the hydrodynamic variable J (r, t) is divergent, then it would be a remarkable situation if its integral over all space f dr J(r, t), is finite. Conversely, if this integral is infinite, we could intepret that divergence as an instability condition. Using the techniques presented earlier [3], we examine these integral for classical and quantum cases and show the following: (1) when the pair potential is non-central, the classical integral in Eq. (1) diverges, violating, it seems, the law of conservation of momentum, a result seen earlier using approximate methods, [1 ], and (2) when the quantum Hamiltonian contains a non-Hermitean contribution, the integral of Eq. (2) diverges as well, a behavior consistent with our earlier exact model result [2]. Thus, we have found exact criteria for the divergence of the current, which we have earlier linked to the onset of turbulence [17]. From the physical point of view, we wish to suggest a "principle of limiting excitation", which may be expressed in the following way: when energy is continuously injected into a many-body system as to produce a runaway situtation, the system reacts by awakening its internal degrees of freedom. We propose that such a generalized principle may cover, in addition, turbulence. For example, when a fluid is externally driven to create a runaway situation as to increase velocity indefinitely, or increase kinetic energy without limit, the system reacts by exciting internal degrees of freedom such as molecular rotation, vibration, or any other degree of freedom which then results in de-excitation, or quantum radiation. Since the distribution of direction of this radiation is statistical, versus completely deterministic, it is accompanied by a stochastic recoil velocity which contributes to the fluctuation of fluid velocities. This view gives a physical origin to the fluctuating velocity added to the gross fluid velocity that is the main feature in the Reynolds treatement of turbulence [10]. This proposal also implies the importance of molecular structure, and as suggested earlier, quantum mechanics. This paper aims to present exact statements on the onset of instability using published techniques [ 1,3-5,18,19]. In pursuit of this goal, we introduce some new formal quantum results, presented in preliminary form in [5], and now generalized to be of direct application to the stability analysis of many-body systems. In Section 2 we quickly review the projection techniques using the integral as a projection operator over all space variables, resulting in a formal expression for the time evolution of the one-particle momentum distribution

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function. We then use the result to calculate the momentum integral in Eq. (1). The instability is manifested by a runaway growth in total momentum for systems with non-central forces. In Section 3, we generalize the quantum version of a projection techniques to arrive at a formal expression for the wave function of one-particle in the company of (N - 1) other particles, introduce a non-Hermitean Hamiltonian, and calculate the current. It is known that such Hamiltonians no longer conserve probability. For this case, we show again the occurrence of an instability. In Section 4, we discuss the results and indicate how one may handle the question of instability in the future.

2. Time evolution of the m o m e n t u m distribution function 2.1. Liouville equation A major (and rarely realized goal) in the description of evolving classical many-body systems is the solution of the Liouville equation OfN -- LfN, Ot

(3)

where f u = f N ( ru, pU), r u = {rl, r2 . . . . rN}, pU = {Pl, P2 . . . . PN} and

(4)

L = Lo + Li, N

LO

=

- Z j=l

(5)

pj " VrJ m

and I

Li = -~ ~ Vrj V (rj - r k ) . j#k

(Vpj - Vpk)

(6)

for a system made up of identical particles of mass m interacting via an arbitrary pair potential V. Formal expressions for f N ( r N, pN, t), involving exponents of the Liouville operator, L, exist, but, except for the trivial case of free particles, practical solutions to the Liouville equation do not exist. In general, one extracts from (3) contracted distribution functions such as the one-particle distribution function, f ( r , p, t). From the one-particle distribution function, one may calculate the density n(r, t) = f dp f ( r , p, t) and the reduced momentum distribution function 4~(p, t) = f dr f ( r , p, t). In this paper we evaluate the reduced momentum distribution function using inverse operators and the method ofiterative projections (MIP). In other papers, the MIP was used to calculate hydrodynamic variables [3] to elucidate the long-standing problem of the divergence of transport coefficients. This section addresses the problem of determining the time evolution of the momentum distribution function for three classes of classical many-body systems: (1) initially non-uniform systems with non-central inter-particle potential; (2) initially uniform systems with non-central inter-particle potentials; (3) initially uniform systems with central inter-particle potentials. In applying MIP and inverse operators to classical many-body systems, we take advantage of several simplifications. Let us define

if

P = Po = - - ~

dr N,

(7)

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where £2 is the volume of the system, Q= Q0= l-P0,

(8)

I = Ii =

(9)

f

dp N - l

P is a projection operator whose effect is to integrate out all particle coordinates. Q is also a projection operator. I is an integration over all but one momentum coordinate. Unlike P and Q, I is not a projection operator. 2.2. Formal solution

Following the standard procedures of the projection technique [7], we can write Ofp Ot = P L f p + P L f o ,

Ofo Ot = Q L f Q + Q L f p

(10)

and the formal solutions t

f p ( t ) = f ( t , 0)fe(0) + f

ds F(t, s ) P L f Q ( s ) ,

o fo(t)

=

(ll)

t

a(t, 0)fo(0) + f ds G(t,s)QLfe(s), o

where F(t, s) = e x p [ P L ( t - s)],

(12)

G(t, s) = e x p [ Q L ( t - s)].

We may iterate [8] Eqs. (11) to obtain a formal series solution for fp: ~ , even

f p = g + h,

where

g =

~

~ , odd

gm and h =

m=O

(13)

)..2 hn, n=O

where t

gm= PLZf

Sl

dsl f

ds2 G(Sl, s2)gm-2(s2),

(14)

go = fe(O),

0

o t

t

s1

h°=eLf

dsl f

o

0

ds2 G(sl, s2)hn-Z(S2),

Sl

hl=PLfdslfds2G(sl,O)fl_p(O). o

(15)

o

In arriving at Eqs. (13)-(15), we used the identities PLP = O,

F(t-

s ) P = P,

P L Q L = P L 2 = P L 2.

(16)

Differentiation of the expressions for gm and hn with time yields t

g m = PL1

ds G(t, s)gm-2(s) 0

(17)

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386 and t

h l,

P L 2 [ ds G(t, s)hn-2(s)

(18)

0 We now perform the following manipulations: (1) differentiate Eqs. (17) and (18) once more with respect to time, (2) use the expressions for g~n and h', to eliminate the integrals, and (3) sum over all m from 0 to infinity, and over all n from 1 to infinity for gm and hn, respectively. We get the second-order differential equations:

~, - PL3( pL2i )-I g, - PoL2ig = g,o - pL3(pL2i )-I go = 0, - PL~(pL2i) lh

--

PoC2h

=

Fl I - -

P L ~ ( P L ~ ) - ' h , = 0.

(19) (20)

We may add these equations to get

~N -- VC~(VC2i )-l~)N -- PoL2~gN = 0 ,

(21)

with the initial conditions 4~N(0) and 4~v (0) = - P L i f N ( O ) . The solution of Eq. (21) can be written as dpN(t ) = exp(Ct/2)[{cosh(tD/2) - sinh(tD/2)D-IC}4aN(O) + s i n h ( t D / 2 ) ( D / 2 ) - l P L i f N ( O ) ] ,

(22)

where formally,

C = PoL{(PoL 2) I

(23)

D = ~ ( P o L ~ ( P o L 2 ) - I ) 2 + 4PoL/2,

(24)

and

¢~N is the N-particle momentum distribution function. Eq. (22) is an exact expression for the N-particle momentum distribution function of a classical many-body system whose component particles are all identical. It is valid for central or non-central potentials, for initially uniform or non-uniform systems. For initially uniform systems with non-central potentials, Eq. (22) simplifies to

ON (t ) = exp( Ct /2) {cosh(t D /2) - sinh (t D /2) D - I C }q~N(0).

(25)

For initially uniform systems with central potentials, Eq. (22) becomes ~U (t) = cosh(t D/2)dpN (0).

(26)

Eqs. (22), (25) and (26) offer qualitatively different results, as we shall demonstrate shortly.

2.3. One-particle momentum distribution function Integrating Eqs. (24)-(26) over all particle momenta but one, and over py and Pz, leaving Px, rewritten simply as p, we get the one-dimensional expressions for each of the above cases:

~p(p, t) = exp \ 2 b Op}

cosh

(;j

4b2 + \ b /

ap

1

~p(p, 0)

A. Murielet al./ PhysicaD 119(1998)381-397 sinh

-

( t // 2

+2 L

387

(C--]2&~ ¢O~O(p'O)

-~ 4b +\b! Opj b

(_~v/4b2 _{_(c]2O ~

Op

(27)

IPoLifN(O)

for an initially non-uniform system with non-central potential, (p(p,t)=exp

(ct 0 ) { (2V/ (c~2L' ~ 2-b0-pP cosh 4 b 2 + \ b ! Op]~O(p,O) -

sinh

-~ 4b + \ b / Op] b

(28)

Op

for an initially uniform system with non-central potential, and

qo(p, t) = cosh (bt ~p ) qg(p,O)

(29)

for an initially uniform system with a central potential. In Eqs. (27)-(29), we use the definitions c-

N(N-1) f ~---~-

and

f

b -- N(N - 1) 2S22

drl dr2

(OV(rl,r2)) 3 ~r[

(30)

dr2 ( OV(rl' r2)'~ 2 \ ~rl ] .

(31)

Observe that c is zero for a central pair potential. The expressions for the one-particle momentum distribution function can be simplified further by using the operator identities

exp(St~---fi) f(p)= f(p+St),

(32)

t

2sinh{(flt/2)(O/Op)}(flt/2)(O/Op) = -2 + f ds ) c°sh t (~~

"

(33)

0

Using the operator identities (32) and (33) yields

l(

~o(p,t)=~o

p+

2

'

t

-t-tfl O )[g(p+Ot+flS + g ( p + ~ 22~ s , O ' )]ds 0

,

(34)

388

A. Muriel et al./Physica D 119 (1998) 381-397

where = c/b,

/5 = V/or2 + 4/52

(35)

and (36)

g(p, O) = Ii PoLifN(O).

Eq. (34) is an expression for the one-particle momentum distribution function in one Cartesian component, valid for all classical many-body systems. When there are no initial correlations, g(p, t) can be written as y(t)Oq)(p, O)/8p, where y(t) is a functional of the density distribution and non-central pair potential, and Eq. (34) is reduced to qg(p, t) = ~q) p +

t,0

p+

t,0

ot + fl t ,

t

0 + f ds×(S)Tp [(p +

+ (p +

O)] .

(37)

0 Eq. (37) is an expression for the one-particle momentum distribution function in one Cartesian component, valid for uncorrelated initial distributions. For initially uniform systems with non-central potentials, Eq. (34) becomes ~0(p,t)=~(p

p+

2

'

For initially uniform systems with central potentials, Eq. (34) reduces to e ( p , t) = ½~(p + bt, o) + ½~(p - bt, 0).

(39)

To prevent runaway solutions, Eq. (39) limits the solution to periodic functions.

2.4. Momentum averages

Integrating p multiplied by Eqs. (39), (38), and (37), in this reverse order, gives us the following expressions for total momentum: (J(t)) = (J(O)),

(40)

for initially uniform systems with central potentials, (J(t)) = (J(O)) - ett/2,

(41)

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389

for initially uniform systems with non-central potentials, and at ( J ( t ) ) = (J(O))

2

2

/ I

t /

t

dsy(s)

(42)

0

for initially uncorrelated non-uniform systems with non-central potentials. Notice that runaway growth in total momentum is manifested in Eqs. (41) and (42) but not in Eq. (40) - an indication that central force-systems are intrinsically more stable than non-central force systems. We also make the following remarks: 1. It appears from the calculations for the momentum distribution function for one Cartesian component that a closed, spatially uniform, non-ideal fluid can never attain equilibrium by itself. However, we do not exclude the possibility of the approach to equilibrium inside containers or under the influence of external fields. 2. The momentum distribution function evolution equations (41) and (42) both exhibit runaway behavior. Among the formulas used in arriving at our expressions for the one-particle momentum distribution function is PoLo = O, a formula which is taken for granted in non-equilibrium statistical mechanics. Strictly speaking, the formula is an identity only when one is dealing with infinite volume systems, or for finite volume systems, when all reduced distribution functions go smoothly to zero at the system's boundary. In macroscopic systems, the formula is often a good assumption as bulk terms greatly exceed surface terms. However, when one deals with issues as ergodicity and long term behavior, one cannot be too sure about neglecting surface terms no matter how small. We cannot assume that the evolution equations will be valid for infinite times without taking into account the contribution of surface terms. As a corollary remark, we cannot simply assume that the time evolution equations we have derived remain valid for mesoscopic systems. 3. The simplified treatment of fluids in this section cannot adequately describe the occurrence of various non-central pair potentials in a system which may be brought about by the existence of different rotational and vibrational states of real molecules in fluids. Furthermore, a realistic discussion of molecules with non-central potentials, such as rod-like molecules with obvious internal degrees of freedom, must include the internal degrees of freedom even in the classical Hamiltonian. We cannot take the divergent behavior of the average momenta as a reflection of the true physical description of the system, but it may be taken as a strong hint that when properly formulated, the Hamiltonian description of non-central forces must include internal degrees of freedom. Once non-central potentials are introduced, it must be considered obligatory that internal degrees of freedom are included. So even in a classical formulation, the Hamiltonian must include these terms, as for example, the rotation of diatomic molecules. That we have not done so, having limited ourselves to the non-vanishing of the integral of the cube of the forces, is a serious limitation of the formalism so far presented. Eventually, we must take into account the distortions due to the interactions. The mathematical treatment must describe distortions induced by hard collisions. Physically, however, these distortions do not last forever. They are treated as permanent in this investigation, and is a true limitation of our classical approach to the introduction of non-central forces. The runaway solutions obtained in this section result from treating a transient situation as permanent. And what about molecules, such as polymers, whose structure is ever-present? An automatic pursuit of the calculation of the average momentum for a system with non-central potential, without introducing internal degrees of freedom, is inadequate. We now emphasize, as a result of our work, that the chosen conventional distribution function description cannot hold for classical particles with complex structure. In finding an alternative way, we must consider a quantum approach, which interestingly enough makes the entire problem more mathematically tractable. 4. It is for this reason that we investigate quantum fluids with molecular structure. However, molecular structure in a quantum description enters our analysis in an indirect way, through the possibility of different excited states, or

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structures, and the obligatory recognition of the decay of excited states to the ground state. This decay towards the ground state can only be described by non-Hermitean Hamiltonians. The use of this non-Hermitean Hamiltonian is not a cheap trick, but demanded by the physics. We now consider such a situation in the next section.

3. Quantum system 3.1. Reduced wave function In a previous paper [5], we derived a formal solution for the one- particle wave function as a function of the initial wave function and the N-particle wave function at initial time: o~

~(t) = ~(0) + Z

( - i t ) J KJ_IpH~N(O) ' j!

(43)

j=l

where K = P HZ( P H) - j , H = Ho + Hi is the Hamiltonian of the system, P = 1-2-u+l f dx N-I is a projection operator and 12 is the volume of the system. In [6], we expanded Eq. (43) to show that it could be expressed as a sum of two parts, the time evolution of the one-particle wave function, identified as a "flow" component, and a more complex contribution, not yet completely understood, from the entire N-body wave function at initial time. We analyzed the solution for the special case when P H / P = 0. In this section, we re-analyze the solution for a general class of pair potentials when this assumption does not hold, expanding the sum term by term, simplifying each of the terms, then regrouping and summing the resulting infinite series, to arrive at an exact, closed, non- perturbative expression for ~ ( t ) = P ~ N (t). First, we observe the following definitions and simplifications: (a) P(Ho + Hi)P = (Ho + a)P, where aP = PI-~P, provided P and H0 commute, which is true for a sum of one-particle Hamiltonians H0; (b) ( P H ) - I P = ]H0(l) + a ] - l P which we will write hereafter as (H0 + a ) - l P with the understanding that H0 occurs only as the one-particle Hamiltonian; (c) we have KP = pH2(pH)-I :

P = P ( H 2 + HoHi + HiHo + Hi2)(Ho

+a)-lP

[Ho(Ho + a ) + P(HiHo + H2)](Ho + a ) - l P

: ] n o ( n o + a ) + P(HiHo + Hi2)]P(Ho + a ) - l P ----[Ho(Ho + a) + P ( n i n o + H2)]P(Ho + a)-l P = [Ho + ( a H o + b)(Ho + a ) - l ] P = (Ho + A)P,

(44)

where A = (aHo + b)(Ho + a) - I and bP -- p H 2 p . With (a), (b) and (c), we can write the first five terms of the series of Eq. (1), to get j = 1:

(-it/h)[HotP(O) + P/-/iqZN(0)],

j =2:

(--it/h)2[H2q*(o) + HoPHitPN(O) + PR~N(O)], 2!

j = 3:

where R = HiHo + H 2.

(--it~h)3 [H3kV(0) + AH2tp(O) + (Hg + AHo)PHiqXN(O) + (Ho + A)PR~N(O)], 3!

A. Murielet al./PhysicaD 119(1998)381-397 j =4:

(-it/h)n[H4ko(O) q_ (HoAHg + AH 3 + A2Hg)~(0) 4! +(H30 + HoAHo + AH~ + A2Ho)PHi~N(O) + (H20+ Hoza + AHo + A2)pRtPN(O)],

j=5:

(--it/h)5 [HoSq~(0) + ( H ~ H 2 + HoAH3 + HoA2Hg

391

5!

+ AHoAH 2 + z12H3 + A3H02)~(0) + (H4 + n2zaHo + Ho,~ZHo + ~ H 3 + AHoAHo + ,63H0)PH~¢'N(0)

+ (H2 + H2A + HoAHo +/40 A2 + AH 2 + AHoA + A2H0 + A3)pRtPN(O)], which allows us to write a regrouping, and resummation

oc (_it/h)j_ 1 (-itlh)Jj! Hd~(O) + Z j! Hd -1PHitPN(O)

q'(t) = j=0

j=l

+ ~ (-itlh)J-e HJo-epRtPN(O)

j~

j=2

4- ~

~

(-it/h)Jj!

HJo-q[SqtP(O)-4-TqPnitl'tN(O)

-4- UqPRq/N(O)]

(45)

q=3 j=q or t

~(t)=exp(-~)tP(o)+

f dslexp(-i~)PHi~u(O) 0

t

Sl

+ f dsif dszexp(-~)PRq'u(O) 0

0 1

+

si dsl

=-

i

Sq I

i

ds2---

o

dsq exp

[Sqt/J(0) q- TqPHitPN(O) nt- UqPRtIIN(O)] ,

(46)

Uq ~- A(Ho -{- A) q-3,

(47)

o

where Sq = A ( H 0 -]- A ) q - 3 n g ,

Tq = A(Ho -t- A ) q - 3 n 0 ,

q = 3..... ~

Eq. (46) is an exact, closed non-perturbative expression of some generality, which we now apply to obtain the current defined by Eq. (2).

3.2. Calculation of current usingfirst two terms For our purpose of this section, we only need to examine the first two terms of Eq. (46) to give t

q.,(t) ~ exp ( - ~ )

qJ (0) q - f d s l e x p ( - - i ~ ) P H i q ' u ( O ) 0

A. Muriel et aL/Physica D 119 (1998) 381-397

392 l

+ f dslexp(-iS@)PHeq/N(O),

(48)

0

Hi+ He

where we have made the change H/ ~ with the intention of making the added Hamiltonian contribution as non-Hermitean. Eq. (48) already contains the contribution of the one-particle, and many-body initial data. Let us put Ho = q/(0) and H = HN (0), also h = 1. Then the current from Eq. (2) becomes

He

t

j(r, t) = Ho*VHo - VHo*Ho +

f

dseiH°(t-S)[H~VPHiHN - VH~)PHiq.'N]

o t

+f

dseiH°(t-s)[H~) VPHeHN -- VH~)PHeHN]

0 t

+ f

dse-iH°(t-s)[H~)VPHi*H*v _ V ~ ) p H * g t ~ ]

o t

+ f dse-iH°(t-s)[q/~VPH*HN_

Vtl/~)PHi*HN]

o t

t

t

t

+f veH;q, +f dseiHoseHTH?vfdse-iHosVeHeHN--fdseiHo'hfdse-iHoSeHeHN dseiHospHyH*vfdse-iHosVPHiHN--fdseiHosvPH*H*vfdse-iHospHiHN

o

o

t

o

o

t

o

t

o

t

o

o

t

+ f

dse-iH°(t-s)[PH*tP~ VHO _ V p / - / e * H ~ 0 ]

0 t

t

t

+fdse'"°sPMYH;fdse-i"osveN-f +fdsei"°'eHy;fdse-'"°'VeHeN-f

dseiH°SVPH*H}

o

o

t

o

t

f

t

0

l

t

dseiH°SVPH*Hf'v f

o

o

dse-iH°s pttiHN

o

dse-iH°S pHeHN"

(49)

o

Consider the term t

t

l

t

f dse;S"oe*kf dse-iSZoVeH,HN--f dse-""OeneHNf dse;SZoVeHYHjv 0

0 0 • = t2[ p H*HTv V P HiHN _ P H e ~ N V P H i* H~v]

and integrate the expressions inside the bracket to give

f

dr[PH*H~IVPHiHN -- PHe~NVPIti*H~vl,

0

(50)

A. Muriel et al./Physica D 119 (1998) 381-397

393

which is zero only if He = He* and Hi =/4/*. If He is anti-Hermitean, the integrand is finite, and Eq. (50) diverges - an instability in time occurs. All other terms containing the non-Hermitean contribution will produce similar divergent terms. From other considerations, it is very unlikely that a full resummation of the infinite series from which the expression comes will yield a finite result. The role of the anti-Hermitean contribution is significant. Hence, a runaway total current will occur. The main conclusion is that a non-Hermitean component in the Hamiltonian produces an instability.

3.3. Calculation using mean free field assumption

We introduce another view of the preceding results. The formal solution of the Schri3dinger equation for the reduced one-particle wave function in Eq. (43) may be written as follows: qJ(t) = ko(O) + K - l [ e x p ( - i K t )

(51)

- 1]PHqJN(O).

The particle current density may be calculated directly from the definition: h j = ~--~(~*VqJ

-

-

~V~*)

(52)

provided, the interaction Hamiltonian Hi represents the average scattering potential in the mean field approximation. We reduce the N- body problem to the problem of propagation of one, essentially free particle in an effective external potential field Heef created by incorporating the interaction Hamiltonian into a non-central external potential field

He. The initial condition ~ (0) = qJ0 may then represent the solution of the time-independent Schrrdinger equation for stationary scattering states. In doing so we have a continuous energy spectrum and we are able to find a formal expression for the current I = f dr j without resorting to Taylor expansion of qJ(t). Substituting ~ (t) directly into the expression for the current, and performing formal differentiation and regrouping of the terms, we may separate the current into three parts: time independent, time dependent oscillatory and the term suspected to be runaway in case the effective external potential field Hamiltonian is a non-Hermitean operator. Thus I = I0 + Iosc(t) + It(t), where I0 = ~mi

dr(qJ*(0)VqJ - o v q / * ( 0 ) ) ,

losc(t) = ~

dr{q/*(O)[VK-l(exp(iK*t) - 1)PHff'N(o) + K-l(exp(-iKt)

- 1)

x ( P V H + PHV)qJN(O)] + [ K * - l ( e x p ( i K * t ) - 1 ) P H * ~ ( 0 ) ] ×, [ V K - l ( e x p ( i K * t ) - 1)PHkON(O) + K - l ( e x p ( - i K t --[~(0)

- 1 ) ( P V H + PHV)ff'N(O)]

+ K - l ( e x p ( - i K t ) - 1)PHqJN(O)][VK *-] (exp(iK*t) - 1 ) P H * ~

+ K *-1 (exp(iK*t) - 1)PVH*q~v(O) + K * - l ( e x p ( i K * t ) - I ) ( P V H * + PH*V)qJN(O)]}

A. Murielet al./Physica D 119 (1998) 381-397

394 and

Ir = ~ h- ~ f

dr{[qJ*(O) + K , _ l ( e x p ( - i K * t ) - 1)PH*q-'~(O)]

x [ - i K - I t V K exp(-iKt)PHON(O) - [q/(0) + K - l ( e x p ( - i K t ) - 1)PHkON(O)]

× [ i K * - l t V K * exp(-iK*t)PH*q~v(O)] } For an anti-Hermitean operator H = - H * , K = - K * the portion of the current containing the wave functions qL tp* becomes:

ht f 2m

dr[qj(0)K_ 1VK e x p ( - i K t ) P H ~ v ( O ) - ~ * ( 0 ) K - 1 V K exp(-iKt)eHqJN(O)],

which is divergent for complex wave functions as t approaches infinity. Could a non-Hermitean component arise spontaneously in a system? Perhaps not so spontaneously, but gradually, as we suggested earlier [3]. This could happen when internal degrees of freedom are excited to produce de- excitation, or radiation that is not self-absorbed by a resonant medium. Thus a system could have a stable regime at low energies, and when the internal states are excited, an unstable regime. A threshold energy, in the form of a minimum excitation energy, is needed.

4. Summary and remarks We observe runaway instability in two cases; (a) when a non-central pair potential is used to study the time evolution of the average or total momentum of an entire system, or (b) when we include a non-Hermitean part in the total Hamiltonian of a quantum system. These results are consistent with our earlier works [ 1,2], except that now the results are formal and not approximate. Physically, we may interpret the new results. We interpret the classical instability as a breakdown ofa probabilistic kinetic description for a system with non-central pair potentials where internal degrees of freedom are ignored. Preserving the general concept of the overall conservation of probability, we detect the onset of instability, which we speculate is identified with turbulence. We earlier proposed [1 ] the following interpretation: when completely spherical atoms collide such that the internal structure of the molecules are unchanged, the flow is laminar, the molecules could just as well be represented by point particles. Thus, continuum hydrodynamics, using the NavierStokes equation, is a valid way of describing laminar flow. When energy is pumped into the system by thermal or mechanical means, the collisions become more energetic, and a threshold energy must be overcome to deform the molecules, rendering the molecules stochastically asymmetric and describable by non-central forces. Our analysis of interacting permanently deformed molecules results in a runaway behavior, an instability expressed as a growth of the total momentum in one Cartesian component. Of course the molecules cannot be permanently deformed, and herein lies the limited applicability of our present analysis to real systems. But an acceptance of stochastic deformation opens the discussion for the need for a quantum description, where instead of talking about deformations, one talks about excited (deformed) molecules. An alternative approach would be to start with a Liouvillian formulation that includes not only the kinetic energy, the pair potential, but also the Hamiltonian that corresponds to the internal degrees of freedom, such as rotations or vibrations. The formalism will certainly be much more challenging, than our approach. But this is a severely

A. Muriel et al. / Physica D 119 (1998) 381-397

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difficult problem. Imagine for example, the daunting task of a many-body formulation of colliding soft microscopic tops. It is for this reason that a quantum mechanical treatment becomes necessary, not only because of the physics, but also because of the mathematics as well. Now in a quantum mechanical approach, we must discuss the collision of molecules in different states, and the resulting transitions, which include the one-way decay towards the ground state. One speaks of lifetimes, which can only be treated by a Hamiltonian that is no longer Hermitean. Herein lies the rationale for entertaining the role of non- Hermitean contributions to the Hamiltonian. We find that not only is probability no longer conserved, as is well-known, the system exhibits qualitatively different behavior that we wish to associate with turbulence. It is curious to observe that many past efforts to solve the kinetic equations for the momentum distribution function have been made, but precisely what is to be done with the results once the problem is solved is not taken to its logical conclusion, like calculating transport coefficients, one rarely ever gets to that point. What we have done instead is to calculate the total momentum of the system, hardly a significant new idea, except that by so doing, we exhibit an instability that threatens to violate conservation of momentum. Figuratively, the threat of such violation is frustrated by the system as it reacts to excite its internal degrees of freedom, which then prevents runaway momentum. This new regime may well be turbulent. Hence our suggestion that there exists a general principle, which we call the "principle of limiting excitation", applied to many-body systems with internal degrees of freedom. We may state the proposed principle in the following way: every time that an external field or energy source forces a physical system to absorb energy and increase kinetic energy and momentum, the system awakens some internal degree of freedom to frustrate the effort. In fluids, we interpret this principle to result in turbulence. A turbulent fluid has a finite, constant total momentum when it is maintained in a "steady state" turbulent condition. Thus, fully turbulent pipe flow has a finite total linear momentum. In the same vein, fully developed, steady state circular Couette flow has a constant angular momentum. These linear and angular momenta may be increased by external forces. The values of momentum and energy simply reach another plateau. The limiting values of the linear momentum, or angular momentum, will be determined by the dissipation of energy, ultimately a quantum phenomenon, as energy is dissipated as non-recoverable radiation. In fluids, both the concept of deformation and dissipation ultimately point to the need to develop a quantum description of deformation or excitation, and, radiation of energy to the surroundings. We suggest that quantum reality, expressed as modifiable structure of molecules, becomes absolutely necessary. Given that radiation is largely an irreversible, one way process of dissipation, the dissipation rate is determined by the decay probabilities of molecular, mostly rotational excited states. These decay probabilities are of the same order of magnitudes for different molecules, hence dissipation rates should eventually appear as independent of bulk characteristics like viscosity, which is after all a macroscopic transport property. Thus, a "prediction" could be made, that dissipation rates, in the energetic, or inertial range, should become independent of viscosity, otherwise known in the field of turbulence as the zeroth law of turbulence. At this point, we feel that to describe this suggested "high-energy hydrodynamics", as we have earlier proposed, a quantum description of turbulence is needed. "High energy" is simply defined, it is the energy needed in any system to excite its internal degrees of freedom. So for a fluid, it is the lowest energy needed to excite molecular rotational states, all within the reach of room temperature excitation or mechanical excitation. For a superfluid (laminar) flow, the energy required is the energy needed to destroy the bosonization of the fluid and render the fluid normal (or turbulent). For a superconducting system, it is the energy needed to destroy the Cooper pairs, rendering a superconducting fluid (laminar) into a normal (turbulent) flow. For noble gases, the gap energy may well be the binding energy of transient molecules. All these phenomena require the concept of an energy gap. There is yet another observation that has to be pointed out, we think for the first time in hydrodynamics. It is well-known that for every symmetry rule in physics, there is a corresponding conservation law, called Noether's theorem. Thus, the isotropy surrounding a particle that possesses a central potential guarantees the conservation of

396

A. Muriel et al./Physica D 119 (1998) 381-397

angular momentum. With the use of non-central potentials, there is no longer conservation of angular momentum. In a fluid of unexcited atoms or molecules, such as molecules that may be modeled as symmetric, or non- polar, angular momentum is conserved. When the molecules are rendered stochastically excited, or deformed, describable classically with a non-central potential, conservation of particle momentum no longer applies. There will be fluctuations of angular momentum on top of the measured time averaged angular momentum. This is an explanation why the vortices formed in hard turbulence are no longer so predictable. Quantum mechanically, lost angular momenta are carried away by radiation, using selection rules well studied in atomic physics. This radiation is not accounted for in most turbulence experiments, representing it as simple, classical dissipation of energy. If one then restricts oneself to conventional fluid motion variables, as most experiments do, no conservation of linear and angular momentum will be observed. In a quantum description, the influence of a non-Hermitean Hamiltonian is interpreted as the need to put a dissipative Hamiltonian, which we tried to illustrate in [2]. Once more energy has been pumped into a system that is already turbulent, turbulence persists and disappears only by dissipation, which we introduce heuristically by a non-Hermitean Hamiltonian. This is the only way we know now to treat a quantum dissipative system. First, energy is pumped into the system by thermal or mechanical agitation, the rate of energy pumping is determined only by the experimental setup, but the dissipation of the energy is determined by the quantum properties of matter. There is a need for both pumping and dissipation, that is why the phenomenon of turbulence occurs only in driven dissipative systems. It is possible to develop a simple kinetic theory of such a quantum system [20], but ultimately, we may need a quantum theory of turbulence to best express the concepts of excitation, energy exchange with fluid motion, and dissipation at the radiative level. A second-quantized development may be necessary.

Acknowledgements One of us, AM, acknowledges the hospitality at the Mason Laboratory of Yale University, and the Institute for Advanced Study in Princeton, where some of the ideas presented in this paper were developed. This research was supported in part by the Natural Science Research Council of the Philippines, the Department of Science and Technology of the Philippines and the ICSC World Laboratory in Lausanne.

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[17] [18] [19] [20]

L. Sirovich (Ed.), New Perspectives in Turbulence, Springer, New York, 1991. A Muriel, Phys. Rev. A 50 (1994) 4286. A. Muriel, P. Esguerra, Phys. Rev. E (1996) 1433. A. Muriel, to be published.

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