On equilibrium radiation and zero-point fluctuations in non-relativistic electron gas
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug On equilibrium radiation and zero-point fluctuations innon-relativistic electron gas
A.M. Ignatov , S.A. Trigger Prokhorov General Physics Institute,Russian Academy of Sciences,Vavilova str. 38, Moscow, 119991 Russia Joint Institute for High Temperatures,Russian Academy of Sciences, Moscow 125412,Izhorskaya str. 13, build. 2, Russiaе-mail: [email protected]
Examination of equilibrium radiation in plasma media shows that the spectralthe energy distribution of such radiation is different from the Planck equilibriumradiation. Using the previously obtained general relations for the spectral energydensity of equilibrium radiation in a system of charged particles, we consider radiationin an electron in the limiting case of an infinitesimal damping. It is shown that zerovacuum fluctuations which are part of the full spectral energy distribution should berenormalized. In this case, the renormalized zero vacuum fluctuations depend on theelectron density. A similar effect should exist in the general case of a quasineutralplasma.
Equilibrium photon energy distribution was established by Max Planck [1,2] and isa fundamental relationship, initiated the development of quantum theory. According toPlanck’s law, the spectral energy density of equilibrium radiation (SEDER), or the so-calledblack body radiation, defined by the expression: e P ( ω ) ≡ dE ( ω ) dω = V ~ π c ω exp( ~ ω/T ) − , (1)where V is the volume in which the radiation is enclosed, T is the temperature of the medium(in energy units) surrounding this volume, c is the speed of light in vacuum. When derivingthe Planck distribution, it is assumed that the photons are in thermodynamic equilibriumwith the surrounding substance, although the explicit interaction of the photons with thewalls of the volume is not considered. Thus, the presence of matter in the cavity wherethe radiation is contained should be assumed. The interaction between photons and matterin the cavity to obtain the Planck type of SEDER should be small enough to ensure theideal photon gas and the absence of significant absorption and damping of electromagneticradiation in the volume of V (the interaction of photons between themselves is extremelyweak). At the same time, the presence of matter and a weak interaction between matter andradiation are necessary for the existence of equilibrium of a photon gas [3]. These conditionsare met with a good degree of accuracy, for example, in a rarefied plasma for any frequency,located far from the frequencies corresponding to the absorption lines of the substance.Planck’s law has been experimentally confirmed many times, for example by pumpinglaser radiation into a cavity in a substance with a small hole, and subsequent observationradiation coming out through this hole. Spectral distribution of radiation emanating froman almost empty cavity in the measured spectral range is close to the Planck’s universalcurve.Studying the explicit effect of the presence of matter on the spectral energy densityof equilibrium radiation began only recently [4-8]. Moreover, in [6] (see also [8]) it wasassumed that the substance is a completely ionized plasma in which the spectrum oftransverse electromagnetic oscillations (for non-relativistic and non-degenerated electrons)is determined by the expression ω = p c k + ω p . In [6], it was supposed that the dampingof oscillations is negligible, and the spatial dispersion determined by the dependence ofthe transverse dielectric permittivity (DP) ε tr ( k, ω ) on the wave vector k is absent. Suchan approach actually corresponds to the Brillouin approximation for the field energy in atransparent medium (see [4, 9] and references therein). The main features of the influence ofplasma on the spectral distribution of radiation revealed in [6] are the absence of radiationfor the frequency range from ω = 0 to ω = ω p and a sharp drop in the total radiation energywith an increase in the characteristic dimensionless parameter ~ ω p /T .A more rigorous field-theoretical approach, based on the non-relativistic approximationfor particles, but taking into account both the temporal and spatial dispersion of thetransverse permittivity, was developed in [7–11]. The results of [8, 9] are based on theapplication of the fluctuation-dissipation theorem (FDT) and the introduction of externalcurrents, and the results [7], [10], [11] on a quantum field approach that does not containexternal sources. The results are close, but lead to some differences. Since the use of third-party sources in the problem of intrinsic SEDER in plasma raises certain questions, belowwe consider the expression for SEDER e ( ω ) in the framework of the quantum field approach[7], [10], in which the vacuum zero oscillations e ( ω ) = V ~ ω / π c in the same form as inthe absence of plasma are excluded from the very beginning e ( ω ) = e P ( ω ) + ∆ e ( ω ) , (2)where ∆ e ( ω ) is completely determined by the transverse permittivity ε tr ( k, ω )∆ e ( ω ) = V ~ ω π c coth (cid:18) ~ ω T (cid:19) (cid:20) c πω Z ∞ dkk Im ε tr ( k, ω ) | ω ε tr ( k, ω ) − c k | − (cid:21) , (3)However, it is not possible to establish the positivity of expression (1) at all frequencies.In this regard, two possible scenarios arise. Either it should be accepted that expression(1) is positive for the exact (but unknown) form of the transverse DP, as well as for somecorrectly chosen approximations for the transverse DP, or zero-point fluctuations should berenormalized taking into account the presence of a plasma medium. In connection with thelatter, one should turn to the full spectral density form e full ( ω ) which, as follows from theexpression for e ( ω ) and (2), (3) equals e full ( ω ) = V ~ ω π c coth (cid:18) ~ ω T (cid:19) c πω Z ∞ dkk Im ε tr ( k, ω )( ω Re ε tr ( k, ω ) − c k ) + ω ( Im ε tr ( k, ω )) (4)and is always positive.Let us consider as an example the simplest case of a collisionless electron system inthe absence of spatial dispersion. At the same time, damping is also absent, since theelectrons have nowhere to transfer the momentum. For such a model, the exact DP ε el ( ω ) (the longitudinal and transverse DP in this case coincide), as is known, equals ε el ( ω ) = 1 − ω p ω (5)Moreover, taking into account the absence of damping and taking ω Im ε tr ( k, ω ) → +0 andforming the δ -function of the argument under the integral ω − e el ( ω ) − c k after integrationwe arrive at the expression e full ( ω ) = V ~ ω π c (cid:18) − ω p ω (cid:19) / coth (cid:18) ~ ω T (cid:19) θ ( ω − ω p ) = V ~ ω π c (cid:18) − ω p ω (cid:19) / θ ( ω − ω p ) + V ~ ω π c ~ ω/T ) − (cid:18) − ω p ω (cid:19) / θ ( ω − ω p ) , (6)where the first term in the second line of (6) can be considered as the renormalized vacuumfluctuations, and the second as the Planck distribution modified by particles.Moreover, the modified vacuum fluctuations, as well as the modified Planck distribution,are already functions of the electron density. The existence of a dependence of zero-pointfluctuations in a plasma medium is in a certain sense similar to the Casimir effect [12], butin the problem under consideration, the distortion of the density of vacuum photons occursnot because of the presence of surrounding walls, but because of the presence of a plasmamedium.In expression (6), the modified Planck distribution is similar to the result obtainedfrom physical considerations in [6], [8] with the difference that in [6] (and in [8] in thecorresponding limit) there is a square root of DP (5) , which characterizes the difference inthe considered field approach and the approximation corresponding to FDT. However, inthe latter case, the main statement of this letter on the occurrence of modified by plasmamedium vacuum oscillations is valid.As easy to show if the zero fluctuations are pick out in a vacuum form, which makes itpossible to consider them as an unobservable "background then the remaining "observable"part of e ( ω ) ≡ e ( ω ) is always negative for large values of ω , which is impossible fromphysical reasons. This means that the renormalization of the vacuum fluctuations in plasmais necessary for the considered model of electron gas.A more extended consideration as well as the manifestation of forces associated withrenormalized zero vacuum oscillations, will be done in a separate publication. [1] M. Planck, Annalen der Phys. vol. p. 719 (1901)[2] M. Planck, Annalen der Physik, vol. p. 553 (1901)[3] L.D. Landau, E.M. Lifshitz, Statistical Physics, part 1 Third Edition, Elsevier, 1980[4] M.L. Levin, S.M. 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