Redistribution of Stochastic Kinetic Energy in Ensembles of Non-Identical Charged Particles

Abstract

The energy exchange processes are studied in the systems of non-identical interacting particles (with different sizes, masses, and charges) with the spatially inhomogeneous distribution of the stochastic kinetic energy sources. To analyze the energy balance in such systems, a theoretical model is considered. The analytical relations are presented that describe the redistribution of the stochastic kinetic energy between two charged particles. The relations proposed are verified by means of simulating numerically the problem for particles with the Coulomb interaction between them. The numerical analysis is performed of the process of the stochastic kinetic energy redistribution in the two-dimensional ensembles with the separated fractions of particles with different sizes and temperatures. The results of this work can be applied to the systems with any type of pair interaction. They can be also used for analyzing the energy exchange processes in inhomogeneous systems that are of interest in plasma physics and in physics of polymers and colloid systems.

1 INTRODUCTION

Studies of the energy exchange processes in inhomogeneous systems of interacting particles are of considerable interest in different fields of scientific and technological research (plasma physics, biology, polymer physics, etc.) [1–5]. A number of topical issues concerns the distinctive features of the physical characteristics of ensembles of non-identical particles with different types of pair interaction, as well as different charges, sizes, dielectric constants, etc. [1–8].

The dusty (complex) plasma is the ionized gas consisting of the micron-sized charged matter particles (dust). Such plasma is widespread in nature and is formed in a number of technological processes [1–3]. In most theoretical and numerical studies of the dusty plasma properties, the systems of identical dust particles are considered, since they are easier to be mathematically described and easier to be understood. However, under real conditions, the dust structures almost never consist of identical particles. Even in laboratory studies, the monodisperse dust particles can have the different charges and/or stochastic kinetic energies, depending on their spatial location [1, 2].

Most laboratory studies of dusty plasmas are carried out using gas discharges of various types [9–12]. Usually, in the centers of the gas discharge chambers, the ion density of plasma slightly exceeds its electron density [13]. Due to this fact, the efficient traps for the negatively charged dust particles form [1, 2]. Under such conditions, the stochastic kinetic energy of dust particles (their “kinetic temperature”) can reach the value of ~0.2–5 eV, which is much higher than the temperature of the surrounding gas. The mechanisms for such “anomalous heating” of dust particles are usually associated with the temporal and/or spatial changes in their charges or locations in the inhomogeneous plasma volume [14–19]. Since the dust particle charge is determined by the local plasma parameters in its vicinity, the power of energy sources feeding dust particle (and, accordingly, its “kinetic temperature”) can considerably vary in space [1, 2]. The inhomogeneous temperature distribution of the surrounding gas, laser radiation used for plasma diagnosing, etc. can also cause inhomogeneous heating of the particle system.

Fluctuations of the dust particle charges, occurring due to the random nature of the ion and electron currents charging these particles, are inherent to plasmas of all types [1, 2]. For the laboratory gas-discharge plasma, the additional stochastic kinetic energy of any dust particle ΔТ_f associated with these fluctuations can be written in the following form [1, 14–16]:

$$\Delta {{T}_{f}} \approx eQ{{\alpha }^{2}}{{E}^{2}}{\text{/}}(2M\nu \upsilon ).$$

(1)

Here Q, М, and ν are the particle charge, mass, and friction coefficient, α ≈ 0.5 is the parameter characterizing the amplitude of the particle charge fluctuations, υ ∝ а_d is the characteristic frequency of these fluctuations for a particle with the radius а_d, and E is the electric field strength in the system under consideration required for the particle to be in the equilibrium position in the fields of the acting forces. Under the microgravity conditions, the ΔТ_f value is determined by the charge fluctuations of the surrounding dust cloud particles, E ≅ E_int ~ Q/d², where d is the mean distance between the dust particles. For the ground-based laboratory experiments, for which the main external n-on‑electric force is the gravity, we can write as follows: E ≅ E_int + E_ext, where E_ext ≅ gM/Q.

The absence of simple theoretical models for describing the energy balance in the systems of non-identical charged particles with the inhomogeneous distribution of heat sources (sources of their stochastic kinetic energy) makes it difficult to analyze the heat transport processes in real systems.

In this paper, we will focus on the heat transport mechanism, which is not associated with the mass transport processes and occurs due to the energy exchange during the stochastic vibrations of individual particles near their equilibrium positions that is impossible in the absence of interaction between the system particles. The distinctive features of energy exchange in the ensembles of non-identical particles are considered under conditions close to those of the laboratory experiments in the gas-discharge plasma. In the case of two interacting particles, the analytical relations are presented, which can be useful for the qualitative analysis of the particle energy exchange processes in the extended systems.

We considered the conditions for the stochastic kinetic energy redistribution in the two-dimensional ensembles with two separated fractions of non-identical particles with different sizes and temperatures. The reasons for the formation of separate fractions of different-sized particles are usually associated with the presence of the thermoforetic forces, ${{F}_{{\text{T}}}} = {{C}_{{\text{T}}}}a_{d}^{2}\nabla {{T}_{n}}$, or the ion drag forces, ${{F}_{{\text{I}}}} = {{C}_{{\text{I}}}}a_{d}^{2}{{u}_{i}}$. Here, a_d is the particle radius, ∇T_n is the temperature gradient in the neutral gas (which is directed towards the decreasing temperature T_n), u_i is the ion drift velocity (which is directed along the external electric field), and the C_T and C_I coefficients depend on the external conditions and parameters of the surrounding plasma [1, 20–22]. The mentioned forces (F_I, F_Т) differently depend on the particle size, as compared to the size-dependence the electric field force acting on the charged dust particle, F_Е ≈ QE ∝ a_d. Due to such directions of the electric fields and the gas temperature gradients, under routine conditions of the gas discharge chambers, both of the mentioned forces (F_I, F_Т) will additionally contribute to the balance of forces that will weaken the effect of the external electric fields F_Е on the dust particle [13]. The larger is the size of the dust particle, the greater will be this contribution. Thus, in the system consisting of two particle fractions, the larger-sized particles will be located at the periphery of the dust cloud.

In the present work, the external electric field was assumed to be linear. In the theoretical and numerical calculations presented in this work, we assumed that the material density ρ is the same for different fractions of particles, ρ₁ = ρ₂, i.e. their mass ratio can be written as follows: M₁/M₂ ∝ (a_d1/a_d2)³. The particle charges were set in accordance with the approximation of the limited orbit motion: Q_i ∝ a_di [1, 2]; and their friction coefficients were set in accordance with the molecular-free approximation: ${{\nu }_{i}} \propto a_{{di}}^{2}{\text{/}}{{M}_{i}}$ [23]. With allowance for the accepted approximations, the ratio of the additional energies (1) acquired by the dust particles with different sizes (a_d1, a_d2) due to charge fluctuations can be estimated as follows: $\Delta T_{f}^{1}{\text{/}}\Delta T_{f}^{2}$ ~ (d₂/d₁)⁴ for E_int ≫ E_ext; or $\Delta T_{f}^{1}{\text{/}}\Delta T_{f}^{2}$ ~ (a_d1/a_d2)² for E_int ≪ E_ext. These estimates are valid, if the gravity is the main non-electric force in the system.

2 TWO PARTICLES (BASIC RELATIONS)

We consider the set of linearized equations of motion describing the deviations of two particles (with charges Q₁₍₂₎, masses M₁₍₂₎ and the pair interaction between them) from their equilibrium positions (ξ₁, ξ₂) in the field of external forces under the action of the random forces F_b1(2), which are the source of the stochastic (thermal) energy of particles:

$${{M}_{1}}\frac{{{{d}^{2}}{{\xi }_{1}}^{{}}}}{{d{{t}^{2}}}} = - v_{1}^{{}}{{M}_{1}}\frac{{d{{\xi }_{1}}}}{{dt}} + {{a}_{1}}{{\xi }_{1}}^{{}} + b{{\xi }_{2}} + F_{{b1}}^{{}},$$

(2a)

$${{M}_{2}}\frac{{{{d}^{2}}{{\xi }_{2}}^{{}}}}{{d{{t}^{2}}}} = - {{v}_{2}}{{M}_{2}}\frac{{d{{\xi }_{2}}}}{{dt}} + {{a}_{2}}{{\xi }_{2}}^{{}} + b{{\xi }_{1}} + F_{{b2}}^{{}},$$

(2b)

where ν₁₍₂₎ are the particle friction coefficients due to their collisions with neutrals of the surrounding gas, and the a₁₍₂₎ and b coefficients depend on the physics and degree of freedom of the problem under consideration.

Let us consider the vertical configuration of particles (see Fig. 1). Then, for their vertical displacements ξ₁₍₂₎= z_1(2), the coefficients are as follows: a₁₍₂₎ = ‒(Q₁₍₂₎β_z – F ') and b = –F  '. For the radial displacements ξ₁₍₂₎= r_1(2), the coefficients are as follows: a₁₍₂₎ = –(Q₁₍₂₎β_r – F/d) and b = F/d. Here, β_r and β_z are the electric field gradients in the radial and vertical directions, respectively, d is the distance between particles, F is the force of interaction between particles, and F ' is its derivative in the z direction. For the Coulomb int-eraction, we can write: F = Q₁Q₂/d²; and F  ' = ‒2Q₁Q₂/d³.

Fig. 1.

Vertical configuration of two particles interacting in the cylindrically symmetric electric field of the trap E = E(z, r).

The problem of the stable configuration of two identical particles interacting with different types of potentials was considered in [24–27]. It was shown that the stability of their vertical configuration is determined by the criterion β_r > β_z; in the opposite case (β_r < β_z), the horizontal particle configuration forms. For two particles with different masses, under conditions of the ground-based experiments (where the gravity cannot be neglected), from the conditions of the balance of forces, we obtain the following relation [8]

$$g({{M}_{1}}{{Q}_{2}}-{{M}_{2}}{{Q}_{1}}) + {{Q}_{1}}{{Q}_{2}}{{b}_{z}}d = {\text{ }}({{Q}_{1}} + {{Q}_{2}})F.$$

(3)

In this case, the condition for the instability development in such a system can be written as follows [8]:

$$\begin{gathered} {{Q}_{1}}{{Q}_{2}}{{\beta }_{r}} < ({{Q}_{1}} + {{Q}_{2}})F{\text{/}}d \\ = {{Q}_{1}}{{Q}_{2}}{{\beta }_{z}} + g({{M}_{1}}{{Q}_{2}}-{{M}_{2}}{{Q}_{1}}){\text{/}}d. \\ \end{gathered} $$

(4)

For M₁₍₂₎ ≡ M and Q₁₍₂₎ ≡ Q, in accordance with the above criterion, this condition takes the form of β_r < β_z. The studies of the conditions (4) for the development of instabilities at different problem parameters are presented in [8].

To find the correlators of the particle velocities and displacements in the set of Eqs. (2a)–(2b), we note that the correlators of the F_b1(2) random force obey the following equations: 〈F_b1〉 = 〈F_b2〉 ≡ 0, 〈F_b1F_b2〉 = 0, 〈F_b1V₂〉 = 〈F_b2V₁〉 ≡ 0, 〈F_b1ξ₂〉 = 〈F_b2ξ₁〉 ≡ 0, 〈F_b1ξ₁〉 = 〈F_b2ξ₂〉 ≡ 0, and 〈F_b1V₂〉 = 〈F_b2V₁〉 ≡ 0. Hereinafter, the angle brackets 〈 〉 denote time averaging at t → ∞. In  the case of particles’ movement along the limited orbits, we can write: 〈ξ₁V₁〉 = 〈ξ₂V₂〉 ≡ 0, and 〈V₁₍₂₎F_b1(2)〉 = $\nu T_{{1\left( 2 \right)}}^{0}$, where $T_{{1\left( 2 \right)}}^{0}$ is the temperature of the heat sources [28–31]. Then the equations for the correlators can be represented in the following form [30, 31]

$${{a}_{{1\left( 2 \right)}}}\langle \xi _{{1(2)}}^{2}\rangle + b\left\langle {{{\xi }_{2}}{{\xi }_{1}}} \right\rangle + T_{{1\left( 2 \right)}}^{0} + \delta {{T}_{{1\left( 2 \right)}}} = 0,$$

(5a)

$$\begin{gathered} - {{\nu }_{{1\left( 2 \right)}}}{{M}_{{1\left( 2 \right)}}}\langle {{\xi }_{{2\left( 1 \right)}}}{{V}_{{1\left( 2 \right)}}}\rangle + {{a}_{{1\left( 2 \right)}}}\langle {{\xi }_{2}}{{\xi }_{1}}\rangle \\ + \;b\langle \xi _{{2\left( 1 \right)}}^{2}\rangle + {{M}_{{1\left( 2 \right)}}}\langle {{V}_{1}}{{V}_{2}}\rangle = {\text{ }}0, \\ \end{gathered} $$

(5b)

$${{\nu }_{{1\left( 2 \right)}}}\delta {{T}_{{1\left( 2 \right)}}} = b\langle {{\xi }_{{2\left( 1 \right)}}}{{V}_{{1\left( 2 \right)}}}\rangle ,$$

(5c)

$$\begin{gathered} - {{\nu }_{1}}{{M}_{1}}\langle {{V}_{1}}{{V}_{2}}\rangle - {{\nu }_{2}}{{M}_{2}}\langle {{V}_{1}}{{V}_{2}}\rangle \\ + {\text{ }}{{a}_{1}}\langle {{\xi }_{1}}{{V}_{2}}\rangle + {{a}_{2}}\langle {{\xi }_{2}}{{V}_{1}}\rangle = 0, \\ \end{gathered} $$

(5d)

$$\langle {{\xi }_{2}}{{V}_{1}}\rangle + \langle {{\xi }_{1}}{{V}_{2}}\rangle = 0.$$

(5e)

Here, ${{T}_{1}} = T_{1}^{0} + \delta {{T}_{1}}$ and ${{T}_{2}} = T_{2}^{0} + \delta {{T}_{2}}$, where T₁ and T₂ are the particle temperatures in the equilibrium state of the system, $T_{1}^{0}$ and $T_{2}^{0}$ are the source energies (which, in the numerical simulations of the problem, correspond to their given/initial temperatures), and δT₁ and δT₂ are the temperature increments characterizing the process of establishing the equilibrium state.

We introduce the notation $\Delta T = T_{2}^{0} - T_{1}^{0}$, and then obtain:

$$\delta {{T}_{1}} = {{b}^{2}}\Delta T{\text{/}}(C{{\nu }_{1}}),$$

(6a)

$$\delta {{T}_{2}} = - {{b}^{2}}\Delta T{\text{/}}(C{{\nu }_{2}}),$$

(6b)

where

$$\begin{gathered} C = {{\left( {{{a}_{2}}{{M}_{1}} - {\text{ }}{{a}_{1}}{{M}_{2}}} \right)}^{2}}{\text{/}}({{\nu }_{1}}{{M}_{1}}{{M}_{2}} + {{\nu }_{2}}{{M}_{2}}{{M}_{1}}){\text{ }} \\ + \;{{b}^{2}}({{\nu }_{1}} + {{\nu }_{2}}){\text{/}}({{\nu }_{1}}{{\nu }_{2}}) - ({{a}_{2}}{{\nu }_{1}}{{M}_{1}} + {\text{ }}{{a}_{1}}{{\nu }_{2}}{{M}_{2}}). \\ \end{gathered} $$

(7)

For two identical particles (Q₁₍₂₎ = Q, M₁₍₂₎= M, ν₁₍₂₎ = ν, and a₁₍₂₎ = а), for the vertical configuration, the solution of set of Eqs. (5a)−(5e) can be written as follows [8]

$$\delta {{T}_{{1\left( 2 \right)}}} = \pm {{b}^{2}}\Delta T{\text{/}}(2\{ {{b}^{2}} - {{\nu }^{2}}Ma\} ).$$

(8)

In this case, δT₁ = –δT₂, and for b² ≫ –ν²Мa, the relation |δT₁₍₂₎| → |ΔT|/2 is true, i.e. energy is uniformly distributed between the particles in the system, and at ν → ∞, we obtain |δT₁₍₂₎| → 0.

Figure 2 shows the δT₁/ΔT and δT₂/ΔT ratios as functions of ν₁/ω, where $\omega = {{(Q_{1}^{2}{\text{/}}{{d}^{3}}{{M}_{1}})}^{{1/2}}}$, which illustrate the energy redistributions between two particles with the Coulomb interaction in the vertical and radial directions at g = 0, β_r/β_z = 4, and $T_{1}^{0} < T_{2}^{0}$.

Fig. 2.

Ratios δT₁/ΔT and δT₂/ΔT as functions of ν₁/ω ratio, calculated for two particles with the Coulomb interaction. Black and grey lines correspond to the vertical and radial directions. The dependences were obtained at β_r/β_z = 4 and $T_{1}^{0} < T_{2}^{0}$ for the following parameters: (1) М₁ = М₂ and d = 0.1 cm; (2) М₂ = 2М₁ and d ≅ 0.105 cm.

In conclusion of this section, we consider two special cases of interaction of the particles with equal masses M₁₍₂₎ = M at $T_{2}^{0} \ne T_{1}^{0}$: (i) ν₁₍₂₎ = ν, a₁ ≠ a₂; and (ii) ν₁ ≠ ν₂, a₁₍₂₎ = а.

In the first case (ν₁₍₂₎ = ν, a₁ ≠ a₂), we obtain the following balance equation:

$$\begin{gathered} \delta {{T}_{{1\left( 2 \right)}}} = \pm {{b}^{2}}\Delta T{\text{/}}(2\{ {{b}^{2}} - {{\nu }^{2}}M({{a}_{1}} + {{a}_{2}}){\text{/}}2 \\ + \;{{({{a}_{1}} - {{a}_{2}})}^{2}}{\text{/}}4\} ). \\ \end{gathered} $$

(9)

In this case, the total energy of the system will be conserved (δT₁ + δT₂ = 0), and at b² ≫ –ν² М(a₁ + a₂), the following asymptotic relation will be true: |δT₁₍₂₎/ΔT | → (2 + (a₁ – a₂)²/2b²)^–1.

We note that, for the dust particles in plasma, the condition a₁ ≠ a₂ can be fulfilled even in the case of their pair interaction, for example, due to the difference in the particle charges, or the difference in the external field gradients at the point of their equilibrium position. Whether these conditions are met depends on the possible spatial changes in the parameters of the surrounding plasma (for example, the changes in the densities and temperature of ions/electrons).

We take into account the force balance equation (3) for two particles at Q₁ ≠ Q₂ and M₁₍₂₎ = M (for g = 0). Then |δT₁₍₂₎/ΔT  | → 0 at (Q₁/Q₂ – Q₂/Q₁)²→ ∞. This is due to the fact that, in the system under consideration, the interaction forces decrease with increasing difference between the Q₁ and Q₂ charges, that is a consequence of the force balance condition (3), which should be satisfied in the equilibrium state of the system. Figure 3 shows the δT_z/ΔT and δT_r/ΔT ratios as functions of the Q₁/Q₂ ratio, which describe the energy redistributions in the vertical and radial directions, respectively. The calculations were performed at β_r/β_z = 4 for the particles with the coulomb interaction spaced by a distance of d ≈ 0.1 cm.

Fig. 3.

Ratios |δT_z /ΔT | and |δT_r /ΔT | as functions of Q₁/Q₂ ratio, calculated for particles with the Coulomb interaction spaced by a distance of d ≈ 0.1 cm.

Now, we consider the second case: M₁₍₂₎ = M, a₁₍₂₎ = а, $T_{2}^{0} \ne T_{1}^{0}$, and ν₁ ≠ ν₂. In this case, the friction coefficients can be unequal, ν₁ ≠ ν₂, for example, due to the presence of the temperature and/or pressure gradients in the surrounding buffer gas. Then, for the particles with equal charges Q₁₍₂₎ ≡ Q, we obtain the balance equations in the following form:

$$\delta {{T}_{1}} = {{b}^{2}}{{\nu }_{2}}\Delta T{\text{/}}({{\nu }_{1}} + {{\nu }_{2}})\{ {{b}^{2}} - {{\nu }_{1}}{{\nu }_{2}}Ma\} {\text{ }},$$

(10a)

$$\delta {{T}_{2}} = - {{b}^{2}}{{\nu }_{1}}\Delta T{\text{/}}({{\nu }_{1}} + {{\nu }_{2}})\{ {{b}^{2}} - {{\nu }_{1}}{{\nu }_{2}}Ma\} {\text{ }}{\text{.}}$$

(10b)

The total energy of the system is not conserved: δT₁ + δT₂ ≠ 0. At b² ≫ –ν₁ν₂ Мa, we obtain: δT₁ + δT₂ → ΔT (ν₂ – ν₁)/(ν₁ + ν₂). Since $\Delta T = T_{2}^{0} - T_{1}^{0}$, this means that, at $T_{2}^{0} > T_{1}^{0}$ and ν₁ > ν₂, the system will lose its energy, and in the case of $T_{2}^{0} < T_{1}^{0}$ and ν₁ > ν₂, it will acquire it. This is due to the fact that, if the friction coefficient of one of the particles in the system under consideration is lesser than that of the other particle, the energy exchange between these particles will be not so active. It is easy to see that, for any friction coefficients ν₁ and ν₂, the kinetic temperature of the entire system will vary in the range from $T_{2}^{0}$ to $T_{1}^{0}$.

For the particles with the Coulomb interaction spaced by a distance of d ≈ 0.1 cm, the |δT₁/ΔT | and |δT₂/ΔT | ratios as functions of the ν₁/ν₂ ratio are shown in Fig. 4 for β_r/β_z = 4.

Fig. 4.

Ratios |δT₁/ΔT | and |δT₂/ΔT | as functions of ν₁/ν₂ ratio, calculated along the vertical direction for particles with the Coulomb interaction spaced by a distance of d ≈ 0.1 cm.

The limiting cases of the last two of the problems under consideration (Q₁/Q₂ ≫ 1 and ν₁/ν₂ ≫ 1) cannot be realized under the laboratory conditions. However, the illustrations shown in Figs. 3 and 4 qualitatively demonstrate how the amount of the redistributed energy depends on the different problem parameters.

3 RESULTS OF NUMERICAL SIMULATIONS AND DISCUSSION

Numerical simulations of the energy exchange processes were performed using the Langevin molecular dynamics method for the particles interacting in the anisotropic electric field of the cylindrically symmetric trap. It was assumed that their interaction can be described by the Coulomb potential. The simulation technique is described in detail in [1, 2].

The simulations were performed for two separate particles and for the two-dimensional ensembles with the separated fractions of particles with different sizes and temperatures (at g = 0). For the particles from different fractions, the M₂/M₁ mass ratios and the $T_{2}^{0}{\text{/}}T_{1}^{0}$ (or $T_{2}^{0}{\text{/}}T_{1}^{0}$) temperature ratios of the heat sources ranged from one to ten. The charges and friction coefficients were set as Q_i ∝ a_di and ${{\nu }_{i}} \propto a_{{di}}^{{ - 1}}$, respectively; i = 1, 2. The ω₁₍₂₎/ν₁₍₂₎ ratio ranged from ~0.7 to ~7, where ${{\omega }_{{1\left( 2 \right)}}} = {\text{ }}{{(Q_{{1\left( 2 \right)}}^{2}{\text{/}}{{d}^{3}}{{M}_{{1\left( 2 \right)}}})}^{{1/2}}}$. Here, for the two-dimensional binary systems, d is the mean distance between particles of a separate fraction. The number of particles in each fraction (N₁, N₂) ranged from 50 to 400.

The temperatures of the heat sources $T_{{1\left( 2 \right)}}^{0}$ ranged from ~0.2 to ~2 eV, and it was set to be equal for all degrees of freedom: $T_{z}^{0} = T_{x}^{0} = T_{y}^{0}$. In the simulations, the initial stochastic kinetic energy (energy of the sources) was redistributed from the “hotter” particles to the “less hot” ones. In all cases, the observed particle velocity distributions were close to the Maxwellian functions. But, in the case of two particles, the particle energies were redistributed non-uniformly over the degrees of freedom: T_z ≠ T_x = T_y.

Some results of numerical simulations for the two-particle system performed at β_r/β_z = 4, ω₁/ν₁ ≈ 1.4 and d ≈ 0.1 cm are presented in Figs. 5–7 together with the analytical problem solutions. Figure 5 shows the velocity distribution functions f₁(V₁) and f₂(V₂) in the vertical and radial directions for two identical particles at $T_{1}^{0}{\text{/}}T_{2}^{0}$ = 10. According to the results presented in Section 2, the total energy of the system under simulations did not change: (T₂ + T₁) = ($T_{2}^{0} + T_{1}^{0}$) and (δT₁ + δT₂) = 0.

Fig. 5.

Velocity distribution functions for the two-particle system, calculated at β_r/β_z = 4 and М₁ = М₂. (a) f₁(V₁), vertical direction, and (b) f₂(V₂), radial direction. Symbols and solid lines correspond to the results of numerical simulations and the Maxwellian functions with the following temperatures, respectively: (1) $T_{1}^{0}$ = 2.08 eV; (2) $T_{1}^{r}$ ≅ 1.87 eV; (3) $T_{1}^{z}$ ≅ 1.46 eV; (4) $T_{2}^{z}$ ≅ 0.83 eV; (5) $T_{2}^{r}$ ≅ 0.416 eV; and (6) $T_{2}^{0}$ = 0.208 eV.

Fig. 6.

Velocity distribution functions, calculated at β_r/β_z = 4 and М₂ = 2М₁. (a) f₁(V₁), vertical direction, and (b) f₂(V₂), radial direction. Symbols and solid lines correspond to the results of numerical simulations and the Maxwellian functions with the following temperatures, respectively: (1) $T_{1}^{0}$ = 0.208 eV; (2) $T_{1}^{r}$ ≅ 0.33 eV; (3) $T_{1}^{z}$ ≅ 0.66 eV; (4) $T_{2}^{z}$ ≅ 1.53 eV; (5) $T_{2}^{r}$ ≅ 1.94 eV; and (6) $T_{2}^{0}$ = 2.08 eV.

Fig. 7.

Velocity distribution functions, calculated at β_r/β_z = 4 and М₂ =2М₁. (a) f₁(V₁), vertical direction, and (b) f₂(V₂), radial direction. Symbols and solid lines correspond to the results of numerical simulations and the Maxwellian functions with the following temperatures, respectively: (1) $T_{1}^{0}$ = 2.08 eV; (2) $T_{1}^{r}$ ≅ 1.93 eV; (3) $T_{1}^{z}$ ≅1.6 eV; (4) $T_{2}^{z}$ ≅ 0.735 eV; (5) $T_{2}^{r}$ ≅ 0.38 eV; and (6) $T_{2}^{0}$ = 0.208 eV.

The functions  f₁(V₁) and  f₂(V₂) are shown in Figs. 6 and 7 for М₂ = 2М₁, Q₂/Q₁ = 2^1/3, and ν₁/ν₂ = 2^1/3 at $T_{2}^{0}{\text{/}}T_{1}^{0}$ = 10 and $T_{1}^{0}{\text{/}}T_{2}^{0}$ = 10, respectively. In both cases, (δT₁ + δT₂) ≠ 0. But, in the case of $T_{2}^{0}{\text{/}}T_{1}^{0}$ = 10 (see Fig. 6), the energy increment was negative (δT₁ + δT₂) < 0, and (T₂ + T₁)/($T_{2}^{0} + T_{1}^{0}$) ≈ 0.97. While, in the case of $T_{1}^{0}{\text{/}}T_{2}^{0}$ = 10 (see Fig. 7), the energy increment was positive (δT₁ + δT₂) > 0, and (T₂ + T₁)/($T_{2}^{0} + T_{1}^{0}$) ≈ 1.02. We also see that |δT₁/δT₂| ≅ ν_2/ν₁ (according to formulas (10a) and (10b)).

Thus, the results of numerical simulations for two particles are consistent with the conclusions presented in Section 2. The  f₁(V₁) and  f₂(V₂) velocity distribution functions completely correspond to the temperatures calculated using the analytical expressions.

Now we consider the two-dimensional ensembles consisting of two separated fractions of particles with different sizes. In the systems under simulations, the particles were affected by the trap electric forces F_Е ∝ a_d and the external forces ${{F}_{{{\text{T(I)}}}}} \propto a_{d}^{2}$ directed oppositely to the F_Е force. Here, we consider these forces only in the first (linear) approximation. Since the energy balance is the main subject of this study, the detailed analysis of the nature of forces acting on particles, as well as the mechanisms and conditions for the particle separation by size are beyond the scope of this work.

The calculations were performed for different ratios |F_T(I)/F_Е|, which ranged from 5 to 25% for the small-sized particles, and from 10 to ~30% for the large-sized ones, respectively, depending on the ratio of sizes of the large and small particles. In this case, the ∇F₁/∇F₂ ratio of gradients of the F₁₍₂₎ total forces acting on particles varied from ~1.05 to ~1.2. In all cases, the considerable separation of the system into the particle fractions was observed, and the large-sized particles occurred to be at the periphery of the two-dimensional cloud (see Fig. 8).

Fig. 8.

Particle arrangement in the two-dimensional system at N = 600 (N₁ = 200, N₂ = 400). White and grey symbols correspond to the particles with masses М₁ and М₂ = 2М₁, respectively.

The results of numerical simulations shown in Figs. 9 and 10 were obtained under conditions of the microgravity for the particles with the parameters close to the experimental ones [21, 22]: М₂/М₁ = 8, Q₂/Q₁ = 2, ν₁/ν₂ = 2, $T_{1}^{0} + T_{2}^{0}$ = 4, and the ratio of the force gradients was ∇F₁/∇F₂ ~ 1.1. The mean distances between particles of the large-sized and small-sized fractions were d ≈ 0.07 cm and d ≈ 0.035 cm, respectively. We used the following numerical values of ratios: ω₁/ν₁ ~ 6.8 and ω₂/ν₂ ~ 3.4, where ω₁₍₂₎ = ${{(Q_{{1\left( 2 \right)}}^{2}{\text{/}}{{d}^{3}}{{M}_{{1\left( 2 \right)}}})}^{{1/2}}}$. The probabilities f(r) of finding p-articles with different masses М₁₍₂₎ at distances r from the trap center are shown in Fig. 9, and the radial dependences of their kinetic temperatures Т(r) are shown in Fig. 10.

Fig. 9.

Probabilities f (r) of finding particles with different masses М₁ (black line) и М₂ (grey line) at different distances r from the trap center (М₂/М₁ = 8 and Т₁/Т₂ = 4).

Fig. 10.

Ratios of the kinetic temperatures $T(r){\text{/}}T_{2}^{0}$ as functions of the distance r from the trap center, calculated for two particles with different masses. White and grey symbols correspond to the particles with masses М₁ (the ${{T}_{1}}(r){\text{/}}T_{2}^{0}$ ratio) and М₂ (the ${{T}_{2}}(r){\text{/}}T_{2}^{0}$ ratio), respectively (М₂/М₁ = 8, $T_{1}^{0}{\text{/}}T_{2}^{0}$ = 4, and ν₁/ν₂ = 2). Solid black line shows the set particle temperatures and the boundary between the fractions.

The results of studying the energy balance in the particle system consisting of two fractions are shown in Figs. 11–14 for the following parameters: М₂/М₁ = 2, Q₂/Q₁ = 2^1/3, ∇F₁/∇F₂ ~ 1.1, and ν₁/ν₂ = 2^1/3. The friction coefficients were: ν₁ = 5 s^–1 (ω₁/ν₁ ~ 5.6, ω₂/ν₂ ~ 4.5) and ν₁ = 10 s^–1 (ω₁/ν₁ ~ 2.8, ω₂/ν₂ ~ 2.25).

Fig. 11.

The f(r) function for particles with different mas-ses М₁ (black line) and М₂ (grey line) at М₂/М₁ = 2 and Т₁/Т₂ = 10.

Fig. 12.

The $T(r){\text{/}}T_{2}^{0}$ functions for particles with different masses at М₂/М₁ = 2, $T_{1}^{0}{\text{/}}T_{2}^{0}$ = 10, and ν₁/ν₂ = 2^1/3. White and grey symbols correspond to the ${{T}_{1}}(r){\text{/}}T_{2}^{0}$ and ${{T}_{2}}(r){\text{/}}T_{2}^{0}$ functions, respectively. (Circle and diamond symbols correspond to ν₁ = 5 and 10 s^–1, respectively.) Solid black line shows the set particle temperatures and the boundary between the fractions.

Fig. 13.

The f(r) function for particles with different mas-ses М₁ (black line) and М₂ (grey line) at М₂/М₁ = 2 and Т₂/Т₁ = 10.

Fig. 14.

The $T(r){\text{/}}T_{1}^{0}$ function for particles with different masses at М₂/М₁ = 2, $T_{2}^{0}{\text{/}}T_{1}^{0}$ = 10, and ν₁/ν₂ = 2^1/3. White and grey symbols correspond to the ${{T}_{1}}(r){\text{/}}T_{1}^{0}$ and ${{T}_{2}}(r){\text{/}}T_{1}^{0}$ functions, respectively. (Circle and diamond symbols correspond to ν₁ = 5 and 10 s^–1, respectively.) Solid black line shows the set particle temperatures and the boundary between the fractions.

Figures 11 and 12 show the probabilities f (r) of finding particles with different masses at distances r from the trap and the radial dependences of their kinetic temperatures Т(r) at $T_{1}^{0}{\text{/}}T_{2}^{0}$ = 10. And Figs. 13 and 14 show the same dependences at $T_{2}^{0}{\text{/}}T_{1}^{0}$ = 10.

The analysis of the results of numerical simulations of the two-dimensional particle ensembles showed that, along the boundary lines separating two fractions, if the relations |δT₁₍₂₎/T₁₍₂₎| > 3% are true (the energy increments exceed the accuracy of the numerical experiments), then the following relation will be also true: |δT₁(r)/δT₂(r)| = ${\text{|}}({{T}_{1}}(r) - T_{1}^{0}){\text{/}}({{T}_{2}}(r) - T_{2}^{0}){\text{|}}$ ≈ ν₂/ν₁ (see Figs. 10, 12, and 14); where r is the distance to the center of the trap. In this case, the additional energy satisfied the relation δT₁₍₂₎ ∝ ΔT = $T_{2}^{0} - T_{1}^{0}$ and decreased with growing friction coefficients ν₁₍₂₎ (see Figs. 12 and 14).

4 CONCLUSIONS

The energy exchange processes were studied in the dissipative systems consisting of non-identical interacting particles (with different masses, sizes and charges) with inhomogeneous distributions of heat sources and/or any other sources of the stochastic kinetic energy. We considered the theoretical model for analyzing the energy balance, as well as the analytical relations describing the redistribution of the stochastic kinetic energy between two non-identical particles. The relations proposed were verified by performing numerical simulations of the problem for the two-particle systems with the Coulomb interaction.

The numerical analysis was performed of the energy redistribution processes in the two-dimensional ensembles of non-identical particles consisting of two separate fractions. Such ensembles can be formed in the trap electric fields (F_Е ∝ a_d) under the effect of additional forces proportional to the square of the particle radius (${{F}_{{{\text{T(I)}}}}} \propto a_{d}^{2}$), such as, for example, the thermophoretic forces or the ion drag forces.

The results of this work are applicable to the systems with any type of pair (mutual) interactions, and they can be used to analyze the energy exchange processes in inhomogeneous systems that are of interest in plasma physics and in physics of polymers and colloidal systems.