# On true relaxation statistics in gases

###### Abstract

By example of a particle interacting with ideal gas, it is shown that statistics of collisions in statistical mechanics at any degree of the gas rarefaction qualitatively differs from that conjugated with Boltzmann’s hypothetical molecular chaos and kinetic equation. In reality, probability of the particle collisions in itself is random, which results in power-law asymptotic of the particle velocity relaxation. An estimate of its exponent is suggested basing on simple kinematic reasonings.

On true relaxation statistics in gases \sodtitleOn true relaxation statistics in gases \rauthorYu. E. Kuzovlev \sodauthorKuzovlev \PACS05.20.Dd, 05.20.Jj, 05.40.Fb, 05.40.Jc

1. In the classical Boltzmann’s picture of molecular chaos, inter-particle collisions are treated like momentary random events with concentration (number per unit space volume per unit time) assumed in the Boltzmann equation (BE) [1, 2] to be proportional to product of one-particle distribution functions (DF) of participating particles. In reality, collision is process absorbing relative motion of colliding particles. Therefore spatial distribution of concentration of collisions as fictitiously whole momentary events must drift with center-of-mass velocity of their participants. By this reason, as it was noticed in [3], the distribution can not reduce to product of one-particle DFs of the participants. In other words, the Boltzman’s molecular chaos hypothesis is incompatible with exact equations of mechanics and the related BBGKY equations [1, 2]. They do not allow to prescribe to collisions a definite probability.

Importantly, such reasonings are indifferent to value of (mean) number density of particles, , and remain valid in the “Boltzmann-Grad limit” (BGL) when radius of interaction between particles decreases, , while their density increases, , in such manner that their characteristic mean free path length keeps constant, const , although characteristic gas parameter (GP) turns to zero, .

To the same conclusion, - that BE is a model contradicting rigorous statistical mechanics, - we can arrive also in various other ways, - see [4, 5, 6, 7] and references therein. At the same time, in “mathematical physics” formal substantiation of BE is constructed for at least physically rather meaningless limit , , const , with being total number of gas particles [8, 9] (also called “Boltzmann-Grad limit”). Therefore, in view of practical importance of the question [7, 10], additional indications of principal BE’s defects would be useful. All the more in view of today’s use of idealized collision language in description of many-particle processes, e.g. in gas of “partons”, even in the high-energy physics (see [11, 12]).

Notice that our previous “letter” to the present journal [13] was rejected with advise to search another place for our material. It became [5], and here we consider a new approach to the kinetics of molecular Brownian motion, now focusing on statistics of collisions themselves rather than spatial displacements of test particle in homogeneous gas. At that, for more pungency and clarity, we exclude interactions between gas particles (“atoms”) and thus any hydrodynamic effects.

2. Just chosen system is particular case of two-component gas, with one of components being so “rare impurity” that in visible space region it is represented by a single particle. In the Boltzmann kinetics, DF of coordinates and velocity of this particle, , undergoes linearized BE, sometimes called also [1] Boltzmann-Lorentz equation (BLE),

(1) |

with designating Boltzmann-Lorentz kinetic operator (BLO) acting by formula

(2) |

where is DF of velocity of atoms, is the impact parameter vector (), while and are collision integral and operator.

In rigorous statistical mechanics we have to start from the BBGKY hierarchy [1, 2]. For our system it under standard normalization [2, 5] reads [5, 14]

(3) |

where , is momentum of our test “Brownian” particle (BP), is its mass, and momenta and mass of atoms, with being atoms’ coordinates, , , is BP-atom interaction potential (of course, repulsive, short-range and let spherically symmetric), and . If at initial time moment, let , correlations between BP and gas are completely absent, then . Solution to equations (3) is easy obtainable by their direct consecutive time integrations: for

(4) | |||

with designations

3. It may be thought to become simpler when GP is small (in BGL) and therefore probability of coincidence of BP’s encounters with different atoms is small. Seemingly, this allows transformation of each of the integrations in (4) into separate collision integral:

(5) |

so that the sum turns into solution of BLE (1), that is . However, such hopes are mathematically inconsistent.

The matter is that characteristics of some collision in (2), - namely, its relative velocity and rigidly connected to its impact parameter , - in reality appear to be dependent on other collisions. It becomes visible when one rewrites operators in (4) in BP’s frame via :

(6) |

where is reduced mass of BP-atom pair. From this it is clear also that the inter-dependence of collisions is as strong as large is ratio , and vanishes only in the limit of infinitely massive BP. But it is obviously insensitive to GP value since GP does not enter into integrals of (4).

From viewpoint of mechanics, the inter-dependence means merely that chains of BP-atom collisions represented by -th term of (4) are objects of -body problem and as such in general are wittingly irreducible to two-body problem. A volume occupied by these chains in -dimensional phase space of atoms is not equal to product of -dimensional volumes of separately considered pair collisions. Therefore the place of (5) must be occupied by approximation

(7) |

with coefficients reflecting actual phase volumes of collisions treated as links of coherent many-particle events.

4. From viewpoint of randomness of the events and DFs (), the kinematic inter-dependences between collisions look like inter-particle statistical correlations. One can say that they are caused by competition of atoms for their encounter with BP.

Behavior of these correlations in configuration space can be qualitatively estimated [13] by factor , that is they decay with growing BP-atom distances like solid angle in which BP-atom interaction region is seen. Thus, under BGL the correlations do not disappear, but instead are self-similarly rescaled, so that owing to a ball as before receives atoms essentially correlated with BP. To be more precise, literally this concerns separate terms of ’s series expansions analogous to (4). But after their summation the factors become cut-off at .

Hence, correlations of BP with any marked atoms are destroyed far from it and localized in space because of BP’s interaction with the rest of gas [5, 6, 7, 15, 16, 17, 18]. On the other hand, any correlations, once burned in a non-equilibrium process, then never disappear in time, since should conserve, - as required by the “generalized fluctuation-dissipation relations” [6], - information necessary for time reversal of (ensemble of) phase trajectories of the system.

The latter statements can be formulated like a theorem [15] and imply conclusion about fallacy of BLE, that is hypothesis (5), irrespectively to GP smallness. For the proof it is convenient to attract the coming from [13] exact non-equilibrium “dynamical virial relations” (DVR) [5, 14, 15, 16, 17, 18].

5. For BP in ideal gas, the DVR are easy derivable directly from equations (3) [19]. In one of their equivalent forms [17], under our initial conditions, that are

(8) |

We shall apply the first of them to analysis of relaxation of BP’s velocity. With this purpose let us integrate it over BP’s coordinates, thus passing to DFs and . Besides, on the right-hand side perform integration over and pass to . Then multiply both sides by and write out the result in the form

(9) |

where has clear meaning as conditional average value of concentration of atoms at distance from BP under condition that its velocity is known. Correspondingly, is conditional average value of fluctuations in number of atoms surrounding BP [17, 18], i.e. lack or excess of atoms around BP due to their correlations with it.

Notice, however, that the condition “” insufficiently characterizes system’s non-equilibrium as the origin of discussed correlations constantly weakening in the course of ’s relaxation to a stationary distribution (e.g. equilibrium Maxwellian one, if is such). Therefore it is will be better to introduce averaging under condition of realization of a given mode of relaxation, to be represented by some velocity function orthogonal to : . Concretely,

(10) |

where . Then (9) produces

(11) |

The orthogonality helps to guarantee that the denominator in (10) does not turn to zero, although tends to it, during all time of relaxation (i.e. formally ever). As the result, relation (11) reveals transparent connection between a law of relaxation and integral value of accompanying correlations BP-gas.

6. At (infinitely) small GP, or in BGL, stationary distribution of BP’s velocity undoubtedly satisfies BLE: , and is independent on .

Let us suppose that BLE determines also evolution to the stationary state from arbitrary initial one: . Respectively, choose for the role of some of non-stationary eigen modes of BLO, that is solutions of equation with non-zero (hence, negative) eigen-values [1] (or, equivalently, , where is transposed BLO). Then, according to BLE, with exponential function .

Inserting this into DVR (11), we get , which says that number of atoms involved into correlations with BP unrestrictedly grows with time. Thus, we came to conflict with the above underlined finiteness of spatial scales of correlations. Consequently, the exponential relaxation law dictated by BLE (or even its exponential asymptotic) can not be realized in rigorous enough statistical mechanics.

7. Just mentioned contradiction can be removed already within the approximation (7) which suggests relaxation by law and with non-exponential function

(12) |

Now, relation (11) implies ( ). This clearly shows that the expectation of boundedness of correlation integrals, i.e. the quantity here, justifies when possesses asymptotic of power-law type at , for instance, simply with .

If it is so, then it looks natural to associate limit with quantity from [17] which characterized relaxation of BP’s coordinate distribution [7], and its estimate there prompts us that possibly . Next, we want to show that the same number can be extracted in other way from examination of the coefficients .

8. Notice that the element of collision phase volume in BLO (2) in essence is , where means differential of atom’s path relative to BP along their “collision cylinder” [1]. But in a chain of collisions a “cylinder” of anyone of them occurs “broken” and “spread” because of others, thus losing its sense. Therefore, we are forced to define in an inertial reference frame which is common for all particles taking part in the chain, that is in the frame pinned to their center of mass , where . There, a change in position of one or another atom is partly neutralized by induced displacement of the coordinate origin , so that in place of differential we find .

Hence, correspondingly, in place of in the phase volume element such the quantity suggests itself as is necessary for ensuring that in case of single collision (at ) one comes to the value from (2), with . In application to (7) we must assign (full number of atoms in field of vision under integral ). In such way we obtain . , where calibrating multiplier

Of course, this estimate does not pretend to numeric exactness. But this in no way prevents its qualitative validity. At that, it can be improved, taking into account that BLO (2) ignores important kinematic details of collision, as if during it BP was free of a “kickback” and changed velocity stepwise after all. Let us try to compensate a waste from this distortion of mechanics, noticing that its embodiment would result in true BP’s velocity change only if it was supplied by distorting times either BP’s mass, up to effective value , or oppositely atom’s mass, down to . This correction yields .

9. Consequently, the function (12) appears to be

(13) |

where sum of the series is represented by an integral. Clearly, there plays role of (dimensionless) random rate of BP’s velocity relaxation, or random probability of BP’s collisions, while is probability density distribution of . It is not hard to verify that is concentrated on interval where with . This agrees with conclusions obtained in [21] by methods of “quantum field theory in phase space” [14]. The corresponding power-law asymptotic of , i.e. that of velocity relaxation law, confirms the supposition made above.

In more detail, if , then at the relaxation is almost exponential, , while at it shows crossover to power-law “tail”, . If , then the relaxation law everywhere has more or less power-law behavior.

Transforming randomness of the velocity relaxation rate into randomness of rate (diffusivity) of BP’s “Brownian motion”, , , one can find agreement with results of [3, 20] and [5, 6, 7, 15, 17].

Naturally, all that may be addressed to underlying qualitative difference of actual statistics of BP’s collisions from the Poisson type statistics inherent to Boltzmann’s molecular chaos. In reality, in present approximation, according to (13), probability of detection of collisions equals to , where is average number of collisions during observation time. At this probability decreases with time by law instead of exponential one. As the consequence, variance of number of collisions grows as , thus violating the “law of large numbers”.

10. In conclusion, let us comment obvious defect of displayed statistical picture, namely, “quasi-static” character of randomness of relaxation rate, manifesting itself in absence of time argument of . This is usual shortage of approximate approaches to solution of infinite BBGKY hierarchies [7]. Its removal in mathematically more developed theory would lead, in particular, to appearance of effective time dependences of in (7) and transition from quasi-static randomness to flicker fluctuations (1/f noise).

In principle, however, most important thing for us here is difference of from unit at . May be, modern technique of computer calculations gives possibility to verify this at least for the third, “two-collision”, term of (4). Then, from consideration of problem of three bodies only one could obtain more strong evidence of BE’s invalidity than from numeric “molecular dynamics” of very many particles.

To resume, we considered velocity relaxation law for a particle interacting with atoms of ideal gas, and demonstrated that actual kinematics of the interaction, regardless of gas rarefaction, forbids exponential relaxation and instead establishes one possessing power-law asymptotic, with exponent determined by particle to atom mass ratio.

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