Nonequilibrium effective temperature of superfluid vortex tangle

Physics Letters A 359 (2006) 183–186 www.elsevier.com/locate/pla

Nonequilibrium effective temperature of superfluid vortex tangle D. Jou a,∗ , M.S. Mongiovì b a Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain b Dipartimento di Metodi e Modelli Matematici, Università di Palermo, c/o Facoltà di Ingegneria, Viale delle Scienze, 90128 Palermo, Italy

Received 28 March 2006; accepted 9 June 2006 Available online 21 June 2006 Communicated by C.R. Doering

Abstract An effective nonequilibrium temperature in counterflow superfluid turbulence is proposed, as a parameter characterizing a canonical probability distribution function of vortex orientation, and relating the diffusion coefficient of vortex lines to the vortex friction through an Einstein relation. © 2006 Elsevier B.V. All rights reserved. Keywords: Nonequilibrium temperature; Turbulent superfluids

The meaning of temperature out of equilibrium is an open topic of much fundamental and practical interest. Many definitions of effective nonequilibrium temperatures have been proposed in the literature [1–11]. Such definitions are based on extrapolations of equilibrium equations to the nonequilibrium domain. Whereas in the limit leading to equilibrium all the definitions lead to equilibrium temperature, out of equilibrium the different definitions, even when applied to the same system, yield different results. This is not completely surprising, because out of equilibrium there is no energy equipartition and, therefore, the different degrees of freedom will have different temperatures. The aim of this Letter is to examine the concept of nonequilibrium effective temperature in an interesting physical system, the turbulent vortex tangle which is formed in a superfluid under a simultaneous presence of rotation and thermal counterflow [12–15]. The main physical quantities needed to describe the superfluid turbulence are the average vortex line length per unit volume L, briefly called vortex line density, whose dimensions are (length)−2 , and the counterflow velocity V = vn − vs (vn and vs being the velocities of the normal and superfluid components), related to the heat flux q supplied to the system; in the absence

* Corresponding author.

E-mail address: [email protected] (D. Jou). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.06.031

of mass flux it is q = ρT sV (ρ, T and s being the density, the equilibrium temperature and the entropy of helium II). The motivation of the present analysis arises from the analogy between this system and a paramagnetic system. Indeed, it is known that rotation of the superfluid tends to orient the vortex lines along it, whereas the counterflow tends to randomize their orientation. The final result is that the vortex tangle is far from being isotropic, in contrast with what happens in purely thermal counterflow, and that, as a consequence, it exhibits some degree of polarization. Then, the analogy with paramagnetic systems comes immediately in mind. In the latter ones, the external magnetic field H tends to orient along it the microscopic magnetic moments μ  of the particles constituting the system, whereas the thermal agitation, as measured by absolute temperature T , tends to randomize the orientation of the magnetic moments [12–15]. Thus, several authors have used this analogy to characterize the orientation of the tangent unit vector s to the vortex line s(ξ, t) in order to evaluate the polarization of the vortex tangle. Recall that in equilibrium, the canonical distribution quantifies this conflicting tendencies to orientation and disorder through the following probability distribution   −μH cos θ ≡ exp[−x cos θ ], Pr(cos θ ) ∝ exp (1) kB T  and H kB being Boltzmann’s constant, θ the angle between μ and x ≡ μH /kB T .

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Applied to a Langevin system, i.e. a classical system where all orientations are possible, (1) yields [16] 1 cos θ = coth x − . (2) x The analogy suggests that one could try to quantify the ordering and disordering tendencies of rotation (characterized by the angular velocity Ω) and the counterflow (characterized by the relative velocity V related to a heat flux across the system), respectively, by means of an extension of (1). It is true that (1) is only valid in equilibrium situations, in such a way that one should be careful in sticking to such a heuristic idea as a fundamental law. On the other side, information theoretical arguments would yield to (1) if the average values of cos θ  are known. Thus, we will explore this possibility, to ascertain for its degree of consistency. One of the practical problems is what one should take for x in (2) instead of x = μH /kB T . It is obvious that one must take the ordering factor, Ω, in the numerator and the disordering factor V in the denominator. One may do so through a dimensionless combination of Ω, V and κ, where κ = h/m is the quantum of vorticity, having dimensions of (length)2 /time. In particular, one may set Ωκ , (3) V2 where a(T ) is a dimensionless constant which may depend on the actual temperature of the superfluid. Note that such temperature is not expected to play a prominent role in (3), as the disordering of vortices is not produced at all by the equilibrium thermal agitation, but by the effect of the counterflow. Since κ = h/m (h being the Planck constant and m the mass of helium atom), one may also write (3) as: x=a

x = 2a 1 2

hΩ mV 2

.

(4)

Thus, this ratio is related to the Planck quantum of energy corresponding to the frequency of rotation Ω, over the “kinetic energy”, with a velocity characterized by the counterflow velocity rather than the microscopic thermal velocity. Then, we could interpret 1 mV 2 (5) a(T ) as an effective temperature of the tangle. Obviously, this is a nonequilibrium temperature, because it depends on the external parameter V = vn − vs , which is in turn related to the heat q supplied to the system as q = ρT sV. Nonequilibrium temperatures depending on the fluxes are usual, for instance, √ in extended thermodynamics [1,16–20]. In other terms, V / a(T ) could be interpreted as the mean square velocity of the random motion of the vortices with respect to the superfluid under the presence of an average counterflow velocity V . Thus, 1/a(T ) could be interpreted as the disordered fraction of the kinetic energy of the vortex lines with respect to the superfluid. Note in fact that the fraction between the numerator and the denominator of (4), which is the relevant variable appearing in (1), rather than the numerator and the denominator by kB Teff =

themselves, could be multiplied by an arbitrary number N , indicating, for instance, some numbers of helium atoms. In fact, we are considering vortices, rather than single atoms, and a natural choice for N could be, perhaps, the number of atoms per unit length of vortex line. However, the unit length is rather arbitrary too, so that we will take the minimum possible number of atoms; since the quantum of vorticity corresponds to h/m, m being the mass of only an atom, we consider then a single atom in (5). Now we try to compare (5) with an alternative definition of nonequilibrium effective temperature by starting from other grounds. For instance, a definition which is often used in granular matter or nonequilibrium electrical conductors is the Einstein’s relation between the diffusion coefficient D and the friction coefficient ζ as [21] D=

kB T . ζ

(6)

This relation is strictly valid in equilibrium, but it is used, out of equilibrium, as a definition of an effective temperature by means of D and ζ , which may be measured independently of each other. This leads us to consider the phenomenon of vortex diffusion in an inhomogeneous tangle, where the vortex line density differs from point to point. Following [22], we assume for the vortex line density L a balance equation ∂L + ∇ · J = σ, ∂t

(7)

where J and σ are the vortex diffusion flux and the vortex production term, respectively, and for the diffusion flux an equation of the form: J = −D∇L = −ψ(T )κ∇L,

(8)

in fact κ has the same dimension as a diffusion coefficient, namely (length)2 /time and ψ(T ) is a dimensionless function of T (and perhaps of V ). The diffusion of an inhomogeneous tangle was studied numerically by Tsubota et al. [23], who found that the diffusion constant was about 0.1 times the quantum of circulation, at zero temperature. This provides a detailed quantitative estimate of the coefficient in (8); it would be interesting to study it at nonzero temperatures in the future, to have the full dependence of the coefficient on the temperature. To deal with friction, in the simplest terms, we first generalize (8) to non steady state by means of a relaxational equation of the form of Maxwell–Cattaneo equation [17,24–26], for the heat flux, namely τ

∂J + J = −D∇L, ∂t

(9)

where τ is a relaxation time. For high enough frequencies, this equation predicts a crossover from diffusion behavior, ascribed by ∂L = ψκ∇ 2 L + σ, ∂t

(10)

D. Jou, M.S. Mongiovì / Physics Letters A 359 (2006) 183–186

to a wave behavior ascribed by   1 ∂L 1 ∂ ∂ 2 L ψκ 2 = ∇ L+ σL − + σ. 2 τ τ ∂t τ ∂t ∂t

(11)

The value of τ may be directly related to the friction coefficient acting on the vortices if one assumes, for instance, an homogeneous system (∇L = 0) and studies the decay of vortex motion, as described by ∂J 1 (12) = − J. ∂t τ Since J is related to the velocity of the vortex, (12) may be seen as its equation of motion, of the form ζ lζ0 ∂J = − J = − J. (13) ∂t M lm Here, ζ is the friction coefficient of the vortex, M the mass of the vortex, both of which are proportional to the length of the vortex l; thus ζ0 and m are in principle the friction coefficient per unit length and the mass per unit length. Thus, the relaxation time τ turns out to be independent on the length and is given by τ = m/ζ0 = M/ζ ; therefore, it is ζ = M/τ . Now, we introduce this expression for ζ in Einstein’s relation (6) and define an effective temperature as ψ(T )κM . (14) τ As noted in (5), there is ambiguity in the choice of M, as it was in the choice of ζ . As in (5), we choose for M the minimum significative mass, which is the mass of an helium atom, m. However, it seems that we are still far from (5). However, the relaxation time τ may be given, in phenomenological terms, by

kB Teff = Dζ =

ξ(T ) , (15) κL where ξ(T ) is a dimensionless quantity. This has indeed dimensions of time and implies that the higher the vortex density L, the lower the relaxation time. To grasp better this choice beyond the simple dimensional grounds [27], we may recall the well-known Vinen’s equation [28–30] for the dynamics of L in thermal counterflow, namely:

τ=

185

with τdestr identified as 1/βκL, which is indeed the form proposed in (15), as β(T ) is a dimensionless function of the temperature. In fact, comparison of (15) and (18) leads to the relation ξ(T ) = 1/β(T ). Thus, note that we are considering here the decay of the vortex motion as a consequence of collisions with other vortices, rather than by viscous friction of the fluid. This interpretation is only valid in situations where the vortex line density is high enough in such a way that friction resistance along a mean-free path is relatively small as compared with the loss of momentum in the vortex–vortex collisions. Introducing (15) into (14), recalling that we have chosen M = m, we obtain kB Teff =

ψ(T ) 2 κ mL = ψ(T )β(T )κ 2 mL. ξ(T )

(19)

Finally, we must recall that in steady state, in pure thermal counterflow, from (16) we deduce that one has L = γ 2 (T )V 2 /κ 2 , with γ = α/β. Therefore, we get from (19) kB Teff =

ψ(T ) 2 γ (T )mV 2 = ψ(T )β(T )γ 2 (T )mV 2 . ξ(T )

(20)

Thus, we find an expression analogous to (5). There, the function a(T ) was empirical. Eq. (20) allows us to interpret it in terms of the functions ψ(T ), ξ(T ) (or equivalently β(T )) and γ 2 (T ), all of which has a clear independent meaning, from the corresponding definitions (8), (15) and the well known relation γ 2 (T ) = κ 2 L/V 2 . A comparison with the work of Tsubota et al. [14] is in order. These authors did not mention the concept of nonequilibrium temperature, but took relation (2) as a useful way to obtain the polarity of the vortex tangle in terms of V and Ω, namely:   V2 a(T )Ωκ . − cos θ  = coth (21) a(T )Ωκ V2 They saw that a(T ) has the form a(T ) = 22/γ 2 (T ), a relation obtained from numerical simulations, by looking for the value of a(T ) which fit better the relation (21) between the values of Ω, V 2 and cos θ. This, observing Eq. (22), would indicate that ψ(T ) and ξ(T ) are proportional to each other; this seems physically reasonable, because it implies that the longer the relaxation time (15), the higher will be the diffusion coefficient (8). Note that, comparing (20) and (5), one finds

dL = αv V L3/2 − βv κL2 . (16) dt This equation, in simultaneous presence of counterflow and rotation with angular velocity Ω has been generalized in [13] in the following equation:    1/2  V Ω dL 2 = kL φf , − βv κL2 , (17) dt kL kL1/2

Thus both expressions (20) and (5) share the observation that a(T ) depends inversely on γ 2 (T ). An interesting idea set by Tsubota et al. [14] was to interpret ) LR as x = a(T x = γ222 Ωκ 2 LH , where LR would be the vortex line V2

with the function φf quadratic in its variables. In both Eqs. (16) and (17), the first term is a production term and the second one describes vortex destruction; this second term has the form:   L dL = −βκL2 = − , (18) dt destr τdestr

density due to pure rotation LR = 2Ω/κ and LH = γ 2 V 2 /κ 2 the vortex line density corresponding to pure counterflow. It is well known, however, that the actual vortex line density is not additive, but it is lower than LH + LR [31]. Then, it may be emphasized that (16) leads us, in strict sense, to τeff proportional to L, rather than to LH . However, in going from (19) to (20),

a(T ) =

ξ(T ) 1 . ψ(T ) γ 2 (T )

(22)

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we have replaced L with its value LH in stationary pure counterflow. Summarizing, we have proposed as a measure of the nonequilibrium temperature of the superfluid vortex tangle, the quantity (5), with a(T ) expressed by (22). Thus, we feel that this short note, despite not being completely conclusive, suggests some points which could be conceptually and practically useful in the future. We have tried to explore the analogy of the counterflow rotating vortex tangle with paramagnetism beyond the simple fitting procedure, and have tried to understand whether we may learn more physics from it. This interest must be put into the framework of a more general interest in nonequilibrium effective temperatures. To do so beyond a simple analogy, we have explored a different path leading to a plausible effective temperature, namely, from the Einstein’s formula for the diffusion coefficient. Finally, note that the knowledge of a nonequilibrium temperature could help to achieve a better understanding of such topics as vortex transport in inhomogeneous vortex tangles under combined rotation and counterflow, and to deepen into the limits of validity of the analogy between paramagnetic systems and turbulent vortex tangles.

Acknowledgements

We acknowledge the support of the Acción Integrada España–Italia (Grant S2800082F HI2004-0316 of the Spanish Ministry of Science and Technology and grant IT2253 of the Italian MIUR). D.J. acknowledges the financial support from the Dirección General de Investigación of the Spanish Ministry of Education under grant BFM 2003-06033 and of the Direcció General de Recerca of the Generalitat of Catalonia, under grant 2001 SGR-00186 and 2005-SGR-0087. M.S.M. acknowledges the financial support from MIUR under grant PRIN 200517439-003 and “Fondi 60%” of the University of Palermo.

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