Leonardo Ferrari

Heavy ions in light gases in an electric field: II. Time-dependent solutions of the Fokker-Planck equation both in the absence and in the presence of a magnetic field

Abstract Kiharas time-dependent solution of the approximate Boltzmann equation (Fokker-Planck equation) for heavy ions in light gases in static electric and magnetic fields is extended in order to treat also the cases in which the electric field varies in time. However, since Kiharas solution refers only to the particular situation in which the initial ion velocity distribution is a (shifted) Maxwellian distribution at the equilibrium temperature, a new, quite general method to solve the Fokker-Planck equation with arbitrary initial conditions is also presented. This method reduces the problem to the field-free case, and yields, obviously, the extended Kihara solution as particular case. In the framework of the theory based on the Fokker-Planck equation, our present results show that, both in the absence and in the presence of electric and or magnetic fields (and when diffusion phenomena are negligible), an ion velocity distribution which is initially Maxwellian (at any temperature) around any ion mean velocity, maintains its Maxwellian form during the relaxation process. Moreover, stationary and non-stationary ion velocity distributions in static or alternating electric fields and static magnetic fields are also explicitly obtained.

Heavy ions in light gases in an electric field

Abstract The limits of validity of the Kihara solution of the Boltzmann equation for heavy ions in light gases in an electric field are discussed. To this end the conditions are investigated for the passage from the integral Boltzmann equation to its approximate differential form whose exact solution is the Kihara distribution. It is found that, independently of the presence of an external field, only the Rayleigh condition on the mass ratio is required in the case of an ion-neutral maxwellian interaction. On the contrary, in the presence of an electric field, the main assumption allowing the above passage for any other ion-neutral interaction law implies the further condition that the square of the ratio of the ion drift speed to the thermal velocity of the neutrals cannot exceed (in order of magnitude) the neutral-particle-ion mass ratio. By a direct analysis of the ion velocity averages calculated by using the Kihara distribution, it is shown that this condition (together with Rayleighs) constitutes also the limit of validity of the Kihara distribution in all cases of ion-neutral interaction law.

On the velocity relaxation of a Rayleigh gas

A careful analysis of the assumptions and approximations underlying the derivation of the usual kinetic equation for a Rayleigh gas (or a Brownian particle) is performed. The passage from the exact Boltzmann collision operator to its approximate differential form is thus investigated. It is shown that the exact operator can be replaced by the approximate differential one only when proper conditions on the initial heavy-particle velocity distribution are satisfied. From this analysis it follows that the usual kinetic equation is unable to describe the initial stages of the relaxation of an initial δ-function or of a maxwellian distribution at a temperature much lower than — or, also, for non-maxwellian interactions, much higher than — the equilibrium temperature. In any case, the ratio of the initial velocity distribution to the equilibrium one cannot present a fine structure, or too appreciable deviations from a polynomial form of first or second degree in the velocity components, in velocity-space regions which have linear dimensions which are not large compared with the heavy-particle root-mean-square velocity change per elastic collision. Moreover, except that in the Maxwell model, the initial mean energy of the heavy particles cannot too largely exceed the equilibrium value.

On the velocity relaxation of a Rayleigh gas: II. An investigation on reliability and accuracy of the usual kinetic equation☆

A procedure to test reliability and accuracy of the usual (approximate) differential collision operator (and therefore of the usual kinetic equation) for a Rayleigh gas (or Brownian particles) is presented. The procedure is applied, in the hard sphere model, to the particular case in which the initial heavy-particle velocity distribution is Maxwellian at a temperature T0 different from the equilibrium temperature T. It is found that the most severe limitations to the reliability and accuracy of the usual collision operator originate, in this case, from the truncated Taylor expansion of the ratio between the initial velocity distribution and the equilibrium one. In particular, for light-particle-heavy-particle mass ratios equal to 5 X 10-2, 10-2, 5 X 10-3 and 10-3, the values of T0/T can respectively be only on the intervals [0.89, 1.14], [0.67, 1.48], [0.57, 1.74] and [0.31, 3.25], if a reasonably good accuracy of the usual approximate collision operator is desired. Moreover, since these limitations are found independently of the heavy-particle-light-particle interaction law, their validity is not restricted to the hard sphere model.

Heavy (or large) ions in a fluid in an electric field: The fundamental solution of the Fokker–Planck equation and related questions

The fundamental solution of the Fokker–Planck equation for heavy (or large) ions in a fluid in a generally time-varying electric field is presented. The solution procedure makes use of a convenient transformation of the variables (position r and velocity v) which is suggested by the physics of the problem, and which reduces the solution of the complete Fokker–Planck equation to the well-known solution of this equation in the field-free case. The result obtained shows that, at any time t, the “local” ion velocity distribution f(r,v,t) (apart from the normalization factor taking into account the total number of the ions) differs from the product of the ion number density n(r,t) by the “global” ion velocity distribution Φ(v,t), for a corrective factor C(r,v,t) which is generally different from unity also in the absence of the electric field. The fundamental solution of the Fokker–Planck equation is then widely discussed, in the long-time limit, in the cases of a static—and of an alternate—electric field acti...

An improved differential form of the Boltzmann collision operator for a Rayleigh gas (or Brownian particles)

A new approximate differential form of the Boltzmann collision operator for a Rayleigh gas (or Brownian particles) is derived. The calculation procedure retains all the terms of first and second order in the small quantity μ1 = M/(m+M), M and m being the masses of the light particles and of the heavy particles, respectively. The special conditions under which the obtained operator can be compared (and agree) with the different one derived by Wannier for heavy ions in a cold gas, are discussed. Moreover, some properties of the new operator are studied. In particular, it is shown that, contrarily to what happens for the usual first-order approximate (Fokker-Planck) collision operator, the new operator has the Burnett functions as eigenfunctions only in the Maxwell model. Finally, the reliability and accuracy of the new operator are examined and compared with those of the usual Fokker-Planck collision operator. From this discussion, the advantages offered by the new operator in many instances result manifest.

On the velocity relaxation of a Rayleigh gas: III. The relaxation of the heavy-particle velocity averages☆

Abstract The relaxation of the most important velocity averages of the heavy component of a Rayleigh gas is considered. The relaxation process is studied first exactly in the Maxwell model, and, subsequently, by using the approximate equations following from the usual kinetic equation, in the Rayleigh limit. In some cases non-negligible discrepancies appear between the exact and the approximate results found for a Rayleigh gas in which a heavy-particle-light-particle Maxwellian interaction takes place. In particular it is shown that in certain situations the heavy-particle mean-square velocity components relax, in the approximate theory, in an unphysical way, unless an appropriate limitation on the initial heavy-particle mean velocity is imposed.

The evolution of simple velocity average of ions in gases in time-varying electric fields and static magnetic fields

Abstract The general time-dependent expressions of the simplest velocity averages of ions in gases in arbitrary time-varying electric fields and static magnetic fields are obtained under the assumption of an ion-neutral Maxwellian interaction. To this end, the relaxation equations for mean velocity, mean square speed (or mean energy), and mean square velocity components of the ions are deduced from the Boltzmann equation and solved under quite general initial conditions. In this context the general time-dependent expression of the ion temperature is also discussed, and the need to separately consider the equal-mass case (ions in their parent gas) is pointed out. All The results obtained are then properly extended to the more general case of ions in gas mixtures. In Addition, the Rayleigh-gas limit is examined in order to compare the results following from Boltzmann and Fokker-Planck equations. Finally, ions in alternate electric fields and static magnetic fields are considered as a specific example, and comparisons are made with previous results.

Reliability of the usual differential collision operator in describing the relaxation of a heavy-particle flow in a light gas☆

Abstract The reliability of the usual (approximate) differential collision operator for heavy particles in a light gas (Rayleigh gas, brownian particles) is examined when the heavy particles, having initially temperature T 0 and flow (or drift) velocity 0 , relax in a background gas in equilibrium at temperature T . It is found that severe limitations must be imposed on T 0 / T and |〈ν〉 0 |, in order that the usual approximate collision operator can have a fair reliability at the initial time.

On the velocity relaxation of a Rayleigh gas: IV. Remarks on the solution of the usual kinetic equation

Abstract Two methods of solution of the usual (Fokker-Planck) kinetic equation for a Rayleigh gas are discussed. One is that based on the so-called, and well-known, “fundamental solution” of the same equation, while the other profits by the expansion of the heavy-particle velocity distribution in eigenfunctions of the Fokker-Planck collision operator. The particular case is considered in which the initial heavy-particle velocity distribution is Maxwellian (around a given flow velocity) at a temperature generally different from that of equilibrium. Moreover, the inadequacies of the usual Fokker-Planck equation in describing the relaxation processes are individuated and discussed devoting particular attention to situations in which the initial heavy-particle velocity distribution is anisotropic. In this regard, exploiting the circumstance that the eigenfunctions of the Fokker-Planck collision operator are also eigenfunctions of the Boltzmann collision operator in the Maxwell model, an enlightening comparison between our results and those of the exact Boltzmann theory is presented.