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Fluctuation electromagnetic slowing down and heating of a small neutral particle moving in the field of equilibrium background radiation
Physics Letters A 339 (2005) 212–216 www.elsevier.com/locate/pla
Fluctuation electromagnetic slowing down and heating of a small neutral particle moving in the field of equilibrium background radiation G.V. Dedkov ∗ , A.A. Kyasov Nanoscale Physics Group, Kabardino-Balkarian State University, Nalchik 360004, Russian Federation Received 7 February 2005; received in revised form 1 March 2005; accepted 4 March 2005 Available online 31 March 2005 Communicated by P.R. Holland
Abstract For the first time, fluctuation electromagnetic force and heating rate of a small neutral polarizable particle moving with an arbitrary velocity V with respect to the c.m. of the equilibrium background radiation are calculated at R λW , where R and λW are the particle radius and Wien wave length of the radiation. 2005 Published by Elsevier B.V.
Interaction of a small neutral polarizable body with surrounding transparent medium filled by equilibrium background radiation (thermalized photon gas) is the well-known problem of fluctuation electrodynamics [1]. However, correct description of the field structure both close to (at a distance of about or smaller than the wave length and the body dimension) and far from the body (at a distance much larger than the body dimension) turns out to be not a trivial task, especially if the body is uniformly moving at arbitrary velocity V with respect to the c.m. of the radiation, while its dimension is much smaller than the characteristic
* Corresponding author.
E-mail address: [email protected] (G.V. Dedkov). 0375-9601/$ – see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.physleta.2005.03.054
wave length of thermal radiation. In fact, one needs to know both the near field and far field structure of the fluctuating electromagnetic field (EMF), in order to explore the fundamental characteristics of the interaction, such as the force acting on and heating (cooling) rate of the particle in the reference system related with the background. As far as we know, the first attempt of this kind has been done when calculating the corresponding linear in velocity V drag force applied to a small moving particle, in papers [2,3], while the frictional drag on a moving semi infinite half-space being exerted upon from the stationary half-space—in [4]. A simpler momentum transfer technique, which has been implemented in earlier works when computing the net radiation force on a sphere moving through the cosmic radiation [5], is not adequate here. In our case,
G.V. Dedkov, A.A. Kyasov / Physics Letters A 339 (2005) 212–216
the particle radius R satisfies the condition R λW , where λW is the Wien wave length of the background radiation, and therefore, the corresponding coupling between the particle and medium is not provided by propagating waves, as it does in [5]. Also, close to the described situation are the ones concerning the Casimir effect [6], the fluctuation mediated dissipative van der Waals interactions [7–10], and heat exchange between the bodies in rest [11–14] and in relative motion [10,15]. Quite recently, we have developed general relativistic theory of fluctuation electromagnetic interaction between a small neutral particle and semi infinite flat polarizable surface, the former being in stationary motion parallel to the surface [9,10]. The particle and the surface are characterized by different temperatures (T1 and T2 ) and arbitrary dielectric properties. The developed formalism proves to be effective in the case when the particle moves through the background radiation, too, but in contrast with the former one, when a solution is also possible in the limit c → ∞ [10,15], to explore the interaction between a particle and background, the relativistic statement of the problem becomes crucially important from the very beginning. Owing to this, the expressions for the drag forces derived in [2,3] turn out to be insufficiently complete. Consider a spherical particle of radius R with the polarizability α(ω) and temperature T1 (in the rest frame of the particle), moving in x-direction of the Cartesian coordinate system related with stationary background radiation of temperature T2 . The particle may be considered as a point-like fluctuating dipole if its dimension is small compared to the Wien length, kB R max(T1 , T2 ) 1, 2π hc ¯
(1)
where kB and h¯ are Boltzmann’s and Planck’s constants. Following [9,10], the fluctuation electromagnetic force Fx and rate of the particle heating (cooling) ˙ are given by (in the frame of resting background) Q Fx = ∇x (dE + mB) , (2) ˙ + mB) ˙ = (dE ˙ Q , (3) where d, m and E, B are the fluctuating dipole (electric and magnetic) moments and components of the electromagnetic field, including both the spontaneous and induced contributions, the angular brackets denote
213
complete quantum and statistical averaging. According to the general theory of electromagnetic fluctuations [1], all the vectorial quantities in (2), (3) are assumed to be the Heisenberg operators corresponding to the reference frame of background radiation. Making use of the known transition from dielectric halfspace with the dielectric function ε(ω), to vacuum, ε(ω) = 1 + iδ sign(ω), δ → 0 [1], and proceeding in the same way as in [9,10], we straightforwardly get from (2), (3) the following expressions ∞ 1 h¯ γ 4 d cos θ cos θ Fx = − 4 dω ω πc 0 −1 × (1 + β cos θ )2 α ωγ (1 + β cos θ ) hωγ h¯ ω ¯ (1 + β cos θ ) − coth × coth , 2kB T2 2kB T1 (4) ∞ 1 hγ ¯ Q˙ = 3 dω ω4 d cos θ πc 0 −1 × (1 + β cos θ )3 α ωγ (1 + β cos θ ) hωγ h¯ ω ¯ (1 + β cos θ ) , − coth × coth 2kB T2 2kB T1 (5) where β = V /c, γ = (1 − β 2 )−1/2 and doubly primed α(ω) denotes imaginary part of the corresponding function. In the nonrelativistic limit β 1 one may take β = 0 in (5) while keeping linear in β terms in (4). Then we obtain 4h¯ V Fx = 3πc5
∞
dΠ(ω, T1 ) dω ω α (ω) dω 5
0
Π(ω, T1 ) − Π(ω, T2 ) d ωα (ω) , (6) ω dω ∞
4h¯ ˙ Q = − 3 dω ω4 α (ω) Π(ω, T1 ) − Π(ω, T2 ) , πc 0 (7) where −1 h¯ ω −1 Π(ω, T ) = exp . kB T +
214
G.V. Dedkov, A.A. Kyasov / Physics Letters A 339 (2005) 212–216
At T1 = T2 = T we get from (4) the known result [2,3]: 4h¯ V Fx = − 3πc5
∞
dω ω5 α (ω)
dΠ(ω, T ) dω
0
h¯ 2 V 1 = 3πc5 kB T
∞ dω 0
ω5 α (ω) sinh2 ( 2kh¯BωT )
.
(8)
A new striking situation occurs when the particle and background are not in equilibrium. Consider two cases: (i) “hot” particle and “cold” background, T1 = T , T2 = 0 Fx = −
16h¯ V 3πc5
∞
dω ω4 α (ω)Π(ω, T ),
(9)
0
(ii) “cold” particle and “hot” background, T1 = 0, T2 = T 4h¯ V Fx = 3πc5
∞
dω ωα (ω)
d 4 ω Π(ω, T ) . dω
0
(10)
In the cases described by (8) and (9), the fluctuation force turns out to be decelerating (frictional), but Eq. (10) may result in accelerative force, too. One should note that same possibility can be realized when a cold particle is moving parallel to the hot surface [15]. Let the particle has a narrow absorption line of frequency ω0 , then for α (ω) we may write down α (ω) = 12 πω0 R 3 δ(ω − ω0 ), and substituting this expression into (10) yields 2 Rω0 3 h¯ ω02 V Fx = 3 c c2 x4 4x 3 , − × exp(x) − 1 4 sinh2 (x/2) x = h¯ ω0 /kB T . (11) 1)−1
The function f (x) = − − sinh2 (x/2) is shown in Fig. 1. One sees that Fx > 0 at x < 3.92, the maximal value fmax = 2.13 is reached at x = 1.86, while the minimal value fmin = −1.07 is reached at x = 5.7. Interestingly, just one glimpse of 4x 3 (exp(x)
0.25x 4 /
Fig. 1. The function f (x), x = h¯ ω0 /kB T .
Eq. (11) reveals its principal difference from the results [5], where the corresponding drug force proved to be proportional to R 2 /c4 . Moreover, owing to the dependence Fx ∼ R 3 (in (11)), the acceleration of the particle does not depend on its mass and dimension. Consider, as a numerical example, acceleration of a Si n-doped nanoparticle having the absorption frequency ω0 = 0.2 eV, imbedded into the background radiation of temperature 900 K. In this case we get f (x) = 1.7, and it stems from (11) that the time of exponential growth of the velocity by e equals 2.6 year. If a particle has the absorption domain at ω0 ∼ 1 eV, there is nothing for it but the decelerating force to prevail and the involved damping time reduces by about 100 to 1000 times. Due to the cutting effect of the exponential temperature factors in (8)–(10), the damping times corresponding to the absorption processes at higher frequencies will be much larger. Also, too large damping times prove to be characteristic ones for the Ohmic dissipation processes within the particle, well below the Wien frequency (α ∼ R 3 ω/σ , with σ the dc conductivity). In the ultrarelativistic limit, γ 1, at same conditions (α (ω) ∼ δ(ω − ω0 )) from (4), (5) we obtain h¯ ω05 R 3 γ −4 c4 2 2γ
× dx (1 − x) Π(ω0 x/γ , T2 ) − Π(ω0 , T1 ) ,
Fx = −
1/2
(12)
G.V. Dedkov, A.A. Kyasov / Physics Letters A 339 (2005) 212–216
hω5 R 3 γ −4 ˙= ¯ 0 Q 2c3 2 2γ
× dx Π(ω0 x/γ , T2 ) − Π(ω0 , T1 ) .
215
(13)
1/2
At T1 = T , T2 = 0 the particle is slowing down and cooling, while the stopping force does not depend on the velocity: 2h¯ ω02 Rω0 3 Π(ω0 , T ), Fx ≈ − (14) c c 3 2 −2 Rω0 ˙ Q ≈ −h¯ ω0 γ (15) Π(ω0 , T ). c At T1 = T2 = T a situation may be different. So, at h¯ ω0 /(kB T ) 1 the first terms in the integrals (12), (13), depending on Π(ω0 x/γ ), turn out to be small, and the formulas (14) and (15) remain true yet. But at h¯ ω0 /(kB T ) > 1, oppositely in dependence of hω ¯ 0/ (kB T γ ), the contributions from Π(ω0 x/γ ) dominate. In this case the sign of Fx and Q˙ may change and formulas (14) and (15) become invalid. At T1 = 0, T2 = T from (12), (13) we get kB T ω0 γ −4 Rω0 3 f1 (x), Fx = (16) c c 3 ˙ = kB T ω0 γ −4 Rω0 f2 (x), Q (17) c where x = h¯ ω0 /(kB T γ ). The functions f1 (x) and f2 (x) are displayed in Figs. 2, 3. In this case the particle is heated, while the force Fx , as it is seen from Fig. 2(b), is accelerating at x < 1.75 and decelerating at x > 1.75. Also, a numerical analysis of general equations (12), (13) shows that steady state ˙ = 0 and Fx = 0) is possible asymptotically only (Q at T1 → T2 = T and h¯ ω0 /kB T 1. The larger is β, the larger value h¯ ω0 /kB T manifests an approximate equilibrium. Finally, we shall briefly touch upon physical meaning of the obtained results related with the particle acceleration. At first sight, it might seem that this violates the second law of thermodynamics. However, it is not so, because we must properly take account not only mechanical and thermal effects related with the particle, but also the net energy change of the EMF, produced by work of fluctuating field over the particle. The corresponding energy balance is given by
(a)
(b) Fig. 2. (a) The function f1 (x), x = h¯ ω0 /γ kB T . (b) Same as on (a).
Fig. 3. The function f2 (x), x = h¯ ω0 /γ kB T .
−dW/dt = Fx V + dQ/dt [10,15], where −dW/dt is the work performed by EMF. Consider, for example,
216
G.V. Dedkov, A.A. Kyasov / Physics Letters A 339 (2005) 212–216
Eqs. (12), (13) at T1 = T2 = T and hω ¯ 0 /(kB T ) > 1, when the force Fx can be positive and the particle is to be heated. Using (12), (13) yields −
dW dQ = Fx V + dt dt =
hω ¯ 05 R 3 γ −4 c3 2γ ×
2
dx x Π(ω0 x/γ , T ) − Π(ω0 , T ) > 0.
1/2
Therefore, the particle acceleration and heating result from the EMF work, giving rise to decrease of the EMF energy. Moreover, it is worthwhile noting that dynamics of the particle is determined by relativistic equation of motion mcβ˙ = γ −3/2 Fx (m is the particle rest mass), which is to be solved simultaneously with the equation for the particle temperature (in its rest frame). This needs a special consideration.
References [1] E.M. Lifshitz, L.P. Pitaevskii, Statisticheskaya Fizika, Part 2, Fizmatlit, Moscow, 2002. [2] V. Mkrtchian, V.A. Parsegian, R. Podgornik, W.M. Saslow, Phys. Rev. Lett. 91 (2003) 220801. [3] J.R. Zurita Sanchez, J.J. Greffet, L. Novotny, Phys. Rev. A 69 (2004) 022902. [4] G. Barton, Ann. Phys. 245 (1996) 361. [5] G.R. Henry, R.B. Feduniak, J.E. Silver, M.A. Peterson, Phys. Rev. 176 (1968) 1451; P.J.E. Peebles, D.T. Wilkinson, Phys. Rev. 174 (1968) 2168; C.V. Heer, R.H. Kohl, Phys. Rev. 174 (1968) 1611. [6] M. Kardar, R. Golestanian, Rev. Mod. Phys. 71 (1999) 1233. [7] J.B. Pendry, J. Phys.: Condens. Matter 9 (1997) 10301. [8] A.I. Volokitin, B.N.J. Persson, Phys. Rev. B 65 (2002) 115419. [9] A.A. Kyasov, G.V. Dedkov, Nucl. Instrum. Methods B 95 (2002) 247. [10] G.V. Dedkov, A.A. Kyasov, Phys. Solid State 45 (2003) 1815. [11] D. Polder, M. Van Hove, Phys. Rev. B 4 (1971) 3303. [12] M.L. Levin, V.G. Polevoi, S.M. Rytov, Zh. Eksp. Teor. Fiz. 79 (1980) 2087. [13] J.B. Pendry, J. Phys.: Condens. Matter 11 (1999) 6621. [14] A.I. Volokitin, B.N.J. Persson, Phys. Rev. B 63 (2001) 205404. [15] G.V. Dedkov, A.A. Kyasov, Phys. Low.-Dim. Struct. 1–2 (2003) 1.
Fluctuation electromagnetic slowing down and heating of a small neutral particle moving in the field of equilibrium background radiation G.V. Dedkov ∗ , A.A. Kyasov Nanoscale Physics Group, Kabardino-Balkarian State University, Nalchik 360004, Russian Federation Received 7 February 2005; received in revised form 1 March 2005; accepted 4 March 2005 Available online 31 March 2005 Communicated by P.R. Holland
Abstract For the first time, fluctuation electromagnetic force and heating rate of a small neutral polarizable particle moving with an arbitrary velocity V with respect to the c.m. of the equilibrium background radiation are calculated at R λW , where R and λW are the particle radius and Wien wave length of the radiation. 2005 Published by Elsevier B.V.
Interaction of a small neutral polarizable body with surrounding transparent medium filled by equilibrium background radiation (thermalized photon gas) is the well-known problem of fluctuation electrodynamics [1]. However, correct description of the field structure both close to (at a distance of about or smaller than the wave length and the body dimension) and far from the body (at a distance much larger than the body dimension) turns out to be not a trivial task, especially if the body is uniformly moving at arbitrary velocity V with respect to the c.m. of the radiation, while its dimension is much smaller than the characteristic
* Corresponding author.
E-mail address: [email protected] (G.V. Dedkov). 0375-9601/$ – see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.physleta.2005.03.054
wave length of thermal radiation. In fact, one needs to know both the near field and far field structure of the fluctuating electromagnetic field (EMF), in order to explore the fundamental characteristics of the interaction, such as the force acting on and heating (cooling) rate of the particle in the reference system related with the background. As far as we know, the first attempt of this kind has been done when calculating the corresponding linear in velocity V drag force applied to a small moving particle, in papers [2,3], while the frictional drag on a moving semi infinite half-space being exerted upon from the stationary half-space—in [4]. A simpler momentum transfer technique, which has been implemented in earlier works when computing the net radiation force on a sphere moving through the cosmic radiation [5], is not adequate here. In our case,
G.V. Dedkov, A.A. Kyasov / Physics Letters A 339 (2005) 212–216
the particle radius R satisfies the condition R λW , where λW is the Wien wave length of the background radiation, and therefore, the corresponding coupling between the particle and medium is not provided by propagating waves, as it does in [5]. Also, close to the described situation are the ones concerning the Casimir effect [6], the fluctuation mediated dissipative van der Waals interactions [7–10], and heat exchange between the bodies in rest [11–14] and in relative motion [10,15]. Quite recently, we have developed general relativistic theory of fluctuation electromagnetic interaction between a small neutral particle and semi infinite flat polarizable surface, the former being in stationary motion parallel to the surface [9,10]. The particle and the surface are characterized by different temperatures (T1 and T2 ) and arbitrary dielectric properties. The developed formalism proves to be effective in the case when the particle moves through the background radiation, too, but in contrast with the former one, when a solution is also possible in the limit c → ∞ [10,15], to explore the interaction between a particle and background, the relativistic statement of the problem becomes crucially important from the very beginning. Owing to this, the expressions for the drag forces derived in [2,3] turn out to be insufficiently complete. Consider a spherical particle of radius R with the polarizability α(ω) and temperature T1 (in the rest frame of the particle), moving in x-direction of the Cartesian coordinate system related with stationary background radiation of temperature T2 . The particle may be considered as a point-like fluctuating dipole if its dimension is small compared to the Wien length, kB R max(T1 , T2 ) 1, 2π hc ¯
(1)
where kB and h¯ are Boltzmann’s and Planck’s constants. Following [9,10], the fluctuation electromagnetic force Fx and rate of the particle heating (cooling) ˙ are given by (in the frame of resting background) Q Fx = ∇x (dE + mB) , (2) ˙ + mB) ˙ = (dE ˙ Q , (3) where d, m and E, B are the fluctuating dipole (electric and magnetic) moments and components of the electromagnetic field, including both the spontaneous and induced contributions, the angular brackets denote
213
complete quantum and statistical averaging. According to the general theory of electromagnetic fluctuations [1], all the vectorial quantities in (2), (3) are assumed to be the Heisenberg operators corresponding to the reference frame of background radiation. Making use of the known transition from dielectric halfspace with the dielectric function ε(ω), to vacuum, ε(ω) = 1 + iδ sign(ω), δ → 0 [1], and proceeding in the same way as in [9,10], we straightforwardly get from (2), (3) the following expressions ∞ 1 h¯ γ 4 d cos θ cos θ Fx = − 4 dω ω πc 0 −1 × (1 + β cos θ )2 α ωγ (1 + β cos θ ) hωγ h¯ ω ¯ (1 + β cos θ ) − coth × coth , 2kB T2 2kB T1 (4) ∞ 1 hγ ¯ Q˙ = 3 dω ω4 d cos θ πc 0 −1 × (1 + β cos θ )3 α ωγ (1 + β cos θ ) hωγ h¯ ω ¯ (1 + β cos θ ) , − coth × coth 2kB T2 2kB T1 (5) where β = V /c, γ = (1 − β 2 )−1/2 and doubly primed α(ω) denotes imaginary part of the corresponding function. In the nonrelativistic limit β 1 one may take β = 0 in (5) while keeping linear in β terms in (4). Then we obtain 4h¯ V Fx = 3πc5
∞
dΠ(ω, T1 ) dω ω α (ω) dω 5
0
Π(ω, T1 ) − Π(ω, T2 ) d ωα (ω) , (6) ω dω ∞
4h¯ ˙ Q = − 3 dω ω4 α (ω) Π(ω, T1 ) − Π(ω, T2 ) , πc 0 (7) where −1 h¯ ω −1 Π(ω, T ) = exp . kB T +
214
G.V. Dedkov, A.A. Kyasov / Physics Letters A 339 (2005) 212–216
At T1 = T2 = T we get from (4) the known result [2,3]: 4h¯ V Fx = − 3πc5
∞
dω ω5 α (ω)
dΠ(ω, T ) dω
0
h¯ 2 V 1 = 3πc5 kB T
∞ dω 0
ω5 α (ω) sinh2 ( 2kh¯BωT )
.
(8)
A new striking situation occurs when the particle and background are not in equilibrium. Consider two cases: (i) “hot” particle and “cold” background, T1 = T , T2 = 0 Fx = −
16h¯ V 3πc5
∞
dω ω4 α (ω)Π(ω, T ),
(9)
0
(ii) “cold” particle and “hot” background, T1 = 0, T2 = T 4h¯ V Fx = 3πc5
∞
dω ωα (ω)
d 4 ω Π(ω, T ) . dω
0
(10)
In the cases described by (8) and (9), the fluctuation force turns out to be decelerating (frictional), but Eq. (10) may result in accelerative force, too. One should note that same possibility can be realized when a cold particle is moving parallel to the hot surface [15]. Let the particle has a narrow absorption line of frequency ω0 , then for α (ω) we may write down α (ω) = 12 πω0 R 3 δ(ω − ω0 ), and substituting this expression into (10) yields 2 Rω0 3 h¯ ω02 V Fx = 3 c c2 x4 4x 3 , − × exp(x) − 1 4 sinh2 (x/2) x = h¯ ω0 /kB T . (11) 1)−1
The function f (x) = − − sinh2 (x/2) is shown in Fig. 1. One sees that Fx > 0 at x < 3.92, the maximal value fmax = 2.13 is reached at x = 1.86, while the minimal value fmin = −1.07 is reached at x = 5.7. Interestingly, just one glimpse of 4x 3 (exp(x)
0.25x 4 /
Fig. 1. The function f (x), x = h¯ ω0 /kB T .
Eq. (11) reveals its principal difference from the results [5], where the corresponding drug force proved to be proportional to R 2 /c4 . Moreover, owing to the dependence Fx ∼ R 3 (in (11)), the acceleration of the particle does not depend on its mass and dimension. Consider, as a numerical example, acceleration of a Si n-doped nanoparticle having the absorption frequency ω0 = 0.2 eV, imbedded into the background radiation of temperature 900 K. In this case we get f (x) = 1.7, and it stems from (11) that the time of exponential growth of the velocity by e equals 2.6 year. If a particle has the absorption domain at ω0 ∼ 1 eV, there is nothing for it but the decelerating force to prevail and the involved damping time reduces by about 100 to 1000 times. Due to the cutting effect of the exponential temperature factors in (8)–(10), the damping times corresponding to the absorption processes at higher frequencies will be much larger. Also, too large damping times prove to be characteristic ones for the Ohmic dissipation processes within the particle, well below the Wien frequency (α ∼ R 3 ω/σ , with σ the dc conductivity). In the ultrarelativistic limit, γ 1, at same conditions (α (ω) ∼ δ(ω − ω0 )) from (4), (5) we obtain h¯ ω05 R 3 γ −4 c4 2 2γ
× dx (1 − x) Π(ω0 x/γ , T2 ) − Π(ω0 , T1 ) ,
Fx = −
1/2
(12)
G.V. Dedkov, A.A. Kyasov / Physics Letters A 339 (2005) 212–216
hω5 R 3 γ −4 ˙= ¯ 0 Q 2c3 2 2γ
× dx Π(ω0 x/γ , T2 ) − Π(ω0 , T1 ) .
215
(13)
1/2
At T1 = T , T2 = 0 the particle is slowing down and cooling, while the stopping force does not depend on the velocity: 2h¯ ω02 Rω0 3 Π(ω0 , T ), Fx ≈ − (14) c c 3 2 −2 Rω0 ˙ Q ≈ −h¯ ω0 γ (15) Π(ω0 , T ). c At T1 = T2 = T a situation may be different. So, at h¯ ω0 /(kB T ) 1 the first terms in the integrals (12), (13), depending on Π(ω0 x/γ ), turn out to be small, and the formulas (14) and (15) remain true yet. But at h¯ ω0 /(kB T ) > 1, oppositely in dependence of hω ¯ 0/ (kB T γ ), the contributions from Π(ω0 x/γ ) dominate. In this case the sign of Fx and Q˙ may change and formulas (14) and (15) become invalid. At T1 = 0, T2 = T from (12), (13) we get kB T ω0 γ −4 Rω0 3 f1 (x), Fx = (16) c c 3 ˙ = kB T ω0 γ −4 Rω0 f2 (x), Q (17) c where x = h¯ ω0 /(kB T γ ). The functions f1 (x) and f2 (x) are displayed in Figs. 2, 3. In this case the particle is heated, while the force Fx , as it is seen from Fig. 2(b), is accelerating at x < 1.75 and decelerating at x > 1.75. Also, a numerical analysis of general equations (12), (13) shows that steady state ˙ = 0 and Fx = 0) is possible asymptotically only (Q at T1 → T2 = T and h¯ ω0 /kB T 1. The larger is β, the larger value h¯ ω0 /kB T manifests an approximate equilibrium. Finally, we shall briefly touch upon physical meaning of the obtained results related with the particle acceleration. At first sight, it might seem that this violates the second law of thermodynamics. However, it is not so, because we must properly take account not only mechanical and thermal effects related with the particle, but also the net energy change of the EMF, produced by work of fluctuating field over the particle. The corresponding energy balance is given by
(a)
(b) Fig. 2. (a) The function f1 (x), x = h¯ ω0 /γ kB T . (b) Same as on (a).
Fig. 3. The function f2 (x), x = h¯ ω0 /γ kB T .
−dW/dt = Fx V + dQ/dt [10,15], where −dW/dt is the work performed by EMF. Consider, for example,
216
G.V. Dedkov, A.A. Kyasov / Physics Letters A 339 (2005) 212–216
Eqs. (12), (13) at T1 = T2 = T and hω ¯ 0 /(kB T ) > 1, when the force Fx can be positive and the particle is to be heated. Using (12), (13) yields −
dW dQ = Fx V + dt dt =
hω ¯ 05 R 3 γ −4 c3 2γ ×
2
dx x Π(ω0 x/γ , T ) − Π(ω0 , T ) > 0.
1/2
Therefore, the particle acceleration and heating result from the EMF work, giving rise to decrease of the EMF energy. Moreover, it is worthwhile noting that dynamics of the particle is determined by relativistic equation of motion mcβ˙ = γ −3/2 Fx (m is the particle rest mass), which is to be solved simultaneously with the equation for the particle temperature (in its rest frame). This needs a special consideration.
References [1] E.M. Lifshitz, L.P. Pitaevskii, Statisticheskaya Fizika, Part 2, Fizmatlit, Moscow, 2002. [2] V. Mkrtchian, V.A. Parsegian, R. Podgornik, W.M. Saslow, Phys. Rev. Lett. 91 (2003) 220801. [3] J.R. Zurita Sanchez, J.J. Greffet, L. Novotny, Phys. Rev. A 69 (2004) 022902. [4] G. Barton, Ann. Phys. 245 (1996) 361. [5] G.R. Henry, R.B. Feduniak, J.E. Silver, M.A. Peterson, Phys. Rev. 176 (1968) 1451; P.J.E. Peebles, D.T. Wilkinson, Phys. Rev. 174 (1968) 2168; C.V. Heer, R.H. Kohl, Phys. Rev. 174 (1968) 1611. [6] M. Kardar, R. Golestanian, Rev. Mod. Phys. 71 (1999) 1233. [7] J.B. Pendry, J. Phys.: Condens. Matter 9 (1997) 10301. [8] A.I. Volokitin, B.N.J. Persson, Phys. Rev. B 65 (2002) 115419. [9] A.A. Kyasov, G.V. Dedkov, Nucl. Instrum. Methods B 95 (2002) 247. [10] G.V. Dedkov, A.A. Kyasov, Phys. Solid State 45 (2003) 1815. [11] D. Polder, M. Van Hove, Phys. Rev. B 4 (1971) 3303. [12] M.L. Levin, V.G. Polevoi, S.M. Rytov, Zh. Eksp. Teor. Fiz. 79 (1980) 2087. [13] J.B. Pendry, J. Phys.: Condens. Matter 11 (1999) 6621. [14] A.I. Volokitin, B.N.J. Persson, Phys. Rev. B 63 (2001) 205404. [15] G.V. Dedkov, A.A. Kyasov, Phys. Low.-Dim. Struct. 1–2 (2003) 1.