Quantum And Classical Gases In Confined Geometry

The solution of kinetic equation for the quantum and classical gases with diiusion boundary conditions has been obtained for an arbitrary capillary width d-to-mean free path l ratio. This solution allowed us to obtain analytical expressions describing both the stationary states of the gas lling in two containers connected by a capillary and the thermal slippage eeect. The relations obtained have made it possible to observe the transition from the Knudsen regime (d l) to the hydrodynamic one (d l). This results take into account the innuence of statistics and geometry of the capillary as well. Non-equilibrium stationary states of the gas 1] lling in the two vessels connected by the channels with a small ratio of its width d to the mean free path l T (Knudsen eeect) is used for the investigation of the classical kinetics of the high rareeed gas. As far as we know the Knudsen eeect has been investigated for the classical gases experimentally and theoretically in the limit l T d only 1]. The present paper continues the investigations started in 2] and here we shall consider the Knudsen eeect observed in the quantum gas of the impuritons of the superruid mixtures of helium isotopes. Non-equilibrium stationary states of the classical and quantum gases with the arbitrary ratio of l T =d and various geometry is investigated theoretically in this paper. The explicit expressions are obtained for the distribution function of the particles without any restrictions in the statistics both for the cylindrical and plane-parallel capillaries of various cross-section. Proceeding from the expression for the distribution function one can calculate all signiicant values for the gas ow in the capillary. In this way the relations between gradients of the temperature and concentration were obtained. This results made it possible to obtain both the stationarity condition for the gas ow analytically and a explicit expression for the thermal slippage coeecient. The above expressions being essentially diierent for the classical and quantum gases. A conventional device for the quasiparticles osmotic pressure measurements consists of two vessels lled with superruid 3 He-4 He mixtures. This vessels are connected by the Vicor glass superleak 3]. The values of concentration n 3 and temperature T may be diierent in these containers. The state with the constant value of the 4 He chemical potential (4 =const) is reached by the overrowing of the superruid component through the super-leak relatively fast. …

Non-equilibrium stationary states of the gas 1] lling in the two vessels connected by the channels with a small ratio of its width d to the mean free path l T (Knudsen e ect) is used for the investigation of the classical kinetics of the high rare ed gas.As far as we know the Knudsen e ect has been investigated for the classical gases experimentally and theoretically in the limit l T d only 1].
The present paper continues the investigations started in 2] and here we shall consider the Knudsen e ect observed in the quantum gas of the impuritons of the super uid mixtures of helium isotopes.Non-equilibrium stationary states of the classical and quantum gases with the arbitrary ratio of l T =d and various geometry is investigated theoretically in this paper.
The explicit expressions are obtained for the distribution function of the particles without any restrictions in the statistics both for the cylindrical and plane-parallel capillaries of various cross-section.Proceeding from the expression for the distribution function one can calculate all signi cant values for the gas ow in the capillary .In this way the relations between gradients of the temperature and concentration were obtained.This results made it possible to obtain both the stationarity condition for the gas ow analytically and a explicit expression for the thermal slippage coe cient.The above expressions being essentially di erent for the classical and quantum gases.
A conventional device for the quasiparticles osmotic pressure measurements consists of two vessels lled with super uid 3  and experimentally.Otherwise this state is not stationary one and it should be considered since a quasistationary one as the gas of impuritons can ow through the superleak representing a set of channels with the nite crosssection.In consequence of the impuritons over ow a real stationary state is established.In this state the ow velocity of impuritons 3] must be equal to zero.
The condition of stationarity can be written as follows 1]: Z dS Z d? d 3 dp z g = 0: (1) Here dS is the cross-section element, d? is the momentum space element, axis z lies along the ow, p z is z-th component of momentum, 3 is the energy of impuriton; g = f ?f 0 is the nonequilibrium term adding to the local-equilibrium function f 0 .It should be noted that the left-hand part of equation ( 1) represents the macroscopic velocity of impuriton gas which is averaged over the cross-section of a capillary.Relation (1) makes it possible to determine the values of the temperature and concentration in each of the vessels connected by a capillary in the stationary state.This values must be di erent essentially in the quasistationary state.The situation when the inequality l T d is not ful lled can be observed in a number of experiments (see for example 4]).In this connection the theoretical investigations of the Knudsen e ect with an arbitrary ratio of l T =d are of great interest.
To investigate the relation of stationarity (1) which is valid for the capillary of an arbitrary cross-section, one has to solve the kinetic equation for the gas of the quasiparticles lling in this capillary with respective boundary condition.The length of the capillary is supposed to be much greater than all typical lengths (mean free paths, width of the capillary).This assumption allows us to consider only slow changes of pressure and temperature along the z-axis of the capillary and to linearize the kinetic equation with respect to the small deviation g from the local equilibrium distribution function f 0 .
The kinetic equation should be completed with the boundary conditions.The latter describes the type of interaction between the quasiparticles and the capillary walls.In the case of di usive re ection this conditions may be written as g (x; y) 2 S; (p; ñ) > 0] = 0; (3) where S are the points of the capillary surface, and ñ is positive normal to S. Let us consider the ow of quasiparticles gas in a cylindrical capillary of radius R by using the BGK-approximation for the collision integral 5].In this case the kinetic equation gives s dg d cos(' ? ) + s z df 0 dz = ?g+ p z f 0 mP F Z d? p z g: (4) Here we use the cylindrical coordinate system.The location and the momentum of the particles are de ned by the vectors r = r( ; '; z) and p = p(p ; ; p z ) respectively; s = p=m; and P F is the pressure of the ideal 9 Fermi-gas.The general solution of equation ( 4) with di usion boundary condition (3) may be found as the sum g = g + g H : (5) The rst term in (5) de nes the kinetics of the quasiparticles gas in the vicinity of the capillary walls inside the layer of typical width of mean free path.This term represents the exact solution of the Cauchy-problem.s dg d cos(' ? ) + s z df 0 dz + g = 0; (6) g( = R; (p; ñ) > 0) = 0: (7) It respects to the usual -approximation and di usive re ection on the capillary walls.
The second term in (5) (9) Here the term with the exponent de nes the interaction of quasiparticles with the walls.It di ers from zero in the vicinity of the capillary walls.
The expression for the deviation g H is obtained from the equation (8) which describes the motion of the quasiparticles gas at a distant from the capillary walls (hydrodynamical ow).The asymptotic expression for the deviation of g H with respect to the small parameter l T = (l T = p 2T =m) can be written as g H = 5F 3=2 4F 5=2 p z rP F mP F l 2 T 2 ?R 2 ? 4 s 2 ] + p z f 0 mP F (1 ?e ? ) rT; (10) where = cos(' ? ) s + q R 2 ? 2 sin 2 (' ? ) s ; and Fermi-function F = 1 for the cases of classical (T > T F ) and degenerate (T 6 T F ) gases, respectively.
The same calculations for the plane-parallel capillary give g = g + g H ; (13) g = ?sz df 0 dz 1 ?exp ?d j s x j ?x s x : (14) The x-axis is perpendicular to the walls of the capillary and z-axis coincides with the direction of the gradient (1 ?e ? ) rT; (15) where = d=j s x j + x=s x .
Relations (9), ( 10), ( 14), (15) completely de ne the ow of the gas without any restriction on statistics of the gas in both cylindrical and planeparallel capillaries with an arbitrary cross-section.The substitution of this relations in (1) determines, in general case, the relation between the gradients of temperature and concentration that makes the state stationary but non-equilibrium.
The general relations will not be written here since they are too cumbersome.The expressions for the limiting cases of both relatively wide and narrow capillary will be presented.
Proceeding from both the expressions for nonequilibrium term g (8)-( 10), ( 13)-( 15) and stationarity condition (1) we have for the wide plane-parallel capillary (d=l T > 1) and for the cylindrical one (R T =l T > 1): In classical region of temperatures (T > T F ) relations ( 16) and (17) give Quantum and classical gases in con ned geometry 11 If the gas of impuritons becomes degenerate (for the temperatures T 6 T F ), relations ( 16), (17) may be written as Here l F = p 2T F =m. Expressions ( 16)-( 21) show that the constant di er- ence of temperatures T = T 2 ?T 1 at the ends of a wide capillary is, in a long run, compensated by the constant di erence of pressures P = P 2 ?P 1 , so P P 1 l 2 T T d 2 T 1 : In this case the gas of impuritons ows along the temperature gradient direction near the axis of the capillary and it ows in the opposite direction in the vicinity of the capillary walls making the complete mass ow be equal to zero.This result coincide with the result of 1].Now let us consider the stationary non-equilibrium state of 3 He-4 He mixture lling in the vessels connected by a narrow capillary (d=l T 6 1; R=l T 6 1) with constant values of temperature at its ends 2].This situation is similar to the stationary state of the system of the gas lling in the volumes separated by means of the thin walls with the pores of size much less than mean free path of the particle of the gas (Knudsen e ect 1]).Proceeding from condition (1) and expressions ( 8)-( 10), ( 13)-(15) with reference to the limit considered one can nd that the stationary state is determined by the fact that the function must be constant along the capillary axis r ; = 0: In classical range of temperatures (T > T F ) the asymptotic expression for the function with respect to the small parameter a = 2d=l T 6 1 may be written as for the cylindrical one.
In the region of the degeneracy T 6 T F one can obtain = ? 1516 a F P F ln a F + < ?
for the cylindrical one.
It should be noted that there is essential di erence between asymptotic relations (23), (24) for the plane-parallel capillary and for the cylindrical one, respectively.The main term in the asymptotic derivation (23) of the function for the case of the plane-parallel capillary is a ln a.As to the case with the cylindrical capillary the main term in expression ( 24) is linear a.This result corresponds to the well-known formula of Knudsen describing the establishment of the equilibrium state in the above mentioned classical system of the gas lling in two volumes separated by the porous wall.In this case the mechanical equilibrium is determined by the following equality between temperatures and pressures of the gas lling in the vessels P 1 = p T 1 = P 2 = p T 2 : (27) This relations may be considered as a requirement for the function (24) to be constant along the capillary axis .For the plane-parallel capillary the stationarity requires the ful llment of the another condition This di erence is caused by the fact that when the walls of the planeparallel capillary approach each another we have hydrodynamical regime in each plane parallel to walls.When the radius R of the cylindrical capillary tends to zero the hydrodynamical regime is retained along the direction of the capillary axis only.

He- 4
He mixtures.This vessels are connected by the Vicor glass superleak 3].The values of concentration n 3 and temperature T may be di erent in these containers .The state with the constant value of the 4 He chemical potential ( 4 =const) is reached by the over owing of the super uid component through the superleak relatively fast.As we know this state was studied both theoretically c I.N.Adamenko, A.I.Chervanyov, K.E.Nemchenko, 1997 ISSN 0452{9910.Condensed Matter Physics 1997 No 9 (7{12)

From
(11)  is the coe cient of thermal slippage.It was introduced phenomenologically for the classical gases 1].