The Relativistic Theory of Fluctuation Electromagnetic Interactions of Moving Neutral Particles with a Flat Surface G

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1815–1828. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1729–1741. Original Russian Text Copyright © 2003 by Dedkov, Kyasov. The Relativistic Theory of Fluctuation Electromagnetic Interactions of Moving Neutral Particles with a Flat Surface G. V. Dedkov and A. A. Kyasov Kabardino-Balkar State University, ul. Chernyshevskogo 173, Nalchik, 360004 Russia e-mail: [email protected] Received January 28, 2003 Abstract—The problem concerning the fluctuation electromagnetic interaction of a neutral particle moving parallel to the boundary of a semi-infinite homogeneous isotropic medium characterized by a permittivity and permeability dependent on the frequency is considered in terms of the fluctuation electrodynamic theory within the relativistic formalism. It is assumed that, in the general case, the particle and the medium have different tem- peratures. Within the proposed approach, general expressions are derived both for conservative (normal to the boundary of the medium) and nonconservative (tangential) forces of the interaction between the particle and the medium and for the thermal heating (cooling) rate of the particle. In the nonrelativistic limit (c ∞), the derived relationships coincide with nonrelativistic analogs available in the literature. It is demonstrated that the tangential force acting on the particle can be either accelerating or decelerating. There can occur a situation when the hot moving particle will be heated and the cold medium will be cooled. The interaction of the high- conductivity medium with a high-conductivity particle is analyzed numerically. The asymptotics of the radia- tive contributions to the heat flux and the tangential force is investigated. It is shown that the inclusion of the relativistic effects leads to a substantial increase in the tangential force and the heat flux at distances greater than 1 µm (as compared to the nonrelativistic case); however, the corresponding dependences exhibit a mono- tonic decreasing behavior over the entire range of studied distances (from zero to several hundreds of microns). © 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION of the fluctuation force. Dorofeyev et al. [22] investi- The fluctuation electromagnetic interaction between gated a similar problem but restricted their consider- neutral particles and a medium is governed by quantum ation to the case of a nonrelativistic relative velocity V and thermal fluctuations of the polarization and magne- and obtained (to the first order in V) the following dependence of the tangential force on the velocity V: tization inducing fluctuation electromagnetic fields 2 both inside and outside the particles and the medium. Fx ~ V/c , where c is the velocity of light in free space. The fluctuation electromagnetic interaction is associ- The related problem of calculating the tangential ated with conservative van der Waals forces [1–3], con- forces of the fluctuation electromagnetic interaction servative Casimir forces [3–6], tangential (nonconser- between two semi-infinite media separated by a plane vative) forces [7–14], heat exchange effects [15–18], vacuum space has been treated in a number of earlier and a number of other interesting phenomena [6]. works [8–10]. However, the relationships deduced in The purpose of this work is to develop a consistent those works are inconsistent with each other and with relativistic theory for calculating the heating rate and their nonrelativistic analogs at c ∞ [11, 12], the fluctuation force acting on a small-sized neutral because, in this limit, they lead to a zero tangential spherical particle moving parallel to a flat boundary of force within an approximation linear in velocity. It a polarization medium under vacuum. Consideration is should be noted that Pendry [11] and Volokitin and Per- given to the conservative (normal) and nonconservative sson [12] took into account the retardation effects but (tangential) components of the fluctuation electromag- restricted their consideration to the case of nonrelativ- netic interaction force. istic relative motion. After appropriate transformations The current state of the art in the study of the non- of the expression for the tangential stress between relativistic problem was analyzed in our recent review semi-infinitive media, Volokitin and Persson [12] [14]. The results of relativistic investigations were derived a relationship describing the interaction of a briefly outlined in [19, 20]. In the framework of the rig- small-sized particle and a flat surface. orous relativistic approach, the dynamic force of fluctu- Unlike the relationships derived in the aforemen- ation electromagnetic interaction between a small- tioned works, the expressions obtained in our recent sized particle and a surface was calculated for the first studies [19–21] completely coincide with nonrelativis- time in our previous work [21], in which we derived tic formulas at c ∞ not only within an approxima- general formulas for normal and tangential components tion linear in velocity V. As a result, considerable 1063-7834/03/4510-1815 $24.00 © 2003 MAIK “Nauka/Interperiodica” 1816 DEDKOV, KYASOV z' vacuum (Fig. 1). The half-space z ≤ 0 is filled with a ε y' medium characterized by a complex permittivity and K' complex permeability µ dependent on the frequency ω. It is assumed that the particle is nonmagnetic in the V intrinsic coordinate system and possesses an electric z α ω ε ω x' dipole polarizability ( ) and (or) a permittivity 1( ). K y The real and imaginary parts of these functions (and z0 Vacuum their derivatives) are designated by one and two primes, respectively. The laboratory coordinate system K is related to the surface at rest, and the coordinate system x µ ω ε(ω) K' is related to the moving particle. In the coordinate Medium ( ), system K, the z axis is perpendicular to the interface and the x axis coincides with the direction of the particle ӷ velocity V. Under the assumption that z0 R (where Fig. 1. Coordinate systems K and K' related to the surface at R is the characteristic particle size), the particle can be rest and a moving particle. considered a point fluctuating electric dipole with the moment d(t). advances have been achieved in the description of It should be noted that, within the relativistic dynamic fluctuation electromagnetic interactions, even approach, the nonmagnetic particle having only the though the relativistic solution of the problem of inter- electric dipole moment in the coordinate system K' pos- action between a small-sized particle and a surface can- sesses the magnetic dipole moment m(t) in the coordi- not be directly extended to the case of a different geom- nate system K. Therefore, in the coordinate system K, etry and, in particular, to the geometry of two semi-infi- the vectors of the electric and magnetic polarizations nite media (and vice versa). induced by the moving particle can be written in the form Our calculation technique is based on the equations ()δ,,, ()δ()δ()() of classical and fluctuation electrodynamics (including P xyzt = xVt– y zz– 0 d t , (1) spontaneous and induced components of the fields and ()δ,,, ()δ()δ()() currents caused by fluctuations) and the formalism of M xyzt = xVt– y zz– 0 m t . (2) fluctuation dissipative relationships. It should be noted Correspondingly, in the coordinate system K' (in the that no additional simplifying assumptions were made general case), we have in the course of our calculations. ()δ,,, ()δ()δ() () This paper is organized as follows. In Section 2, we P' x' y' z' t' = x' y' z' d' t' , (3) formulate the problem and derive general expressions M'()δx',,,y' z' t' = ()δx' ()δy' ()z' m'()t '. (4) for calculating the forces and thermal effects. In Sec- tion 3, the essence of the method used is demonstrated The coordinates (x, y, z, t) and (x', y', z', t') in both sys- within the nonrelativistic approach (the results obtained tems are related by the Lorentz transformations are necessary for providing deeper insight into the rel- x' ==γ ()x – βct , t' γ ()t – βx/c , ativistic approach). The regular solution of the Maxwell (5) equations for the electromagnetic field components y' ==y, z' z, induced by a moving particle in the vacuum region is where β = V/c and γ = (1 – β2)–1/2. The formulas of the found in Section 4. The relativistic relationships for Lorentz transformations for the moments d(t) and m(t) correlators of the physical quantities and components can be derived from similar formulas for the polariza- of the retarded Green’s functions for radiation in the tion vectors P and M specified by the components par- medium are deduced in Section 5. The final formulas allel and perpendicular to the direction of the particular describing the fluctuation medium and the thermal velocity V [23]: heating rate of the particle are given in Section 6. The results of numerical calculations are presented in Sec- 1 P|| ==P||' , P⊥ γ P⊥' + ---[]VM' , (6) tion 7. The main conclusions are summarized in Sec- c tion 8. Computational details are described in the Appendix. Note that, in this work, we use the Gaussian 1 system of units. M|| ==M||' , M⊥ γ M⊥' – ---[]VP' . (7) c By using relationships (1)–(7) and the identity δ(αx) = 2. FORMULATION OF THE PROBLEM δ(x)/|α| and assuming that m'(t') = 0, we obtain AND BASIC EXPRESSIONS () γ –1 () () ' () () '() Let us consider a small-sized neutral spherical par- dx t = dx t' , dy t = dy t' , dz t = dz t' , ticle that moves parallel to the boundary of a semi-infi- (8) () () β ' () () β '() nite medium at a distance z0 from this boundary under mx t = 0, my t = dy t' , mz t = – dz t' . PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THE RELATIVISTIC THEORY OF FLUCTUATION ELECTROMAGNETIC INTERACTIONS 1817 Note that the variables t and t' in expressions (8) are and the relationship between the volume elements in related by transformation (5).

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