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Mesoscopic Non-Equilibrium Thermodynamics Part 8

Tham khảo tài liệu 'mesoscopic non-equilibrium thermodynamics part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Mesoscopic Non-Equilibrium Thermodynamics Application to Radiative Heat Exchange in Nanostructures 201 Jst m a _Peq. m T1 - Peq.m A 21 where TA_QN ffl T peq. m T 2 3 h 22 with h being the Planck constant and N m T the averaged number of photons in an elementary cell of volume h3 of the phase-space given by the Planck distribution Planck 1959 N m T _1_ exp Am RbT -1 23 Moreover the factor 2 in Eq. 22 comes from the polarization of photons. The stationary current 21 provides us with the flow of photons. Since each photon carries an amount of energy equal to hm the heat flow Q12 follows from the sum of all the contributions as Q12 I hrnJst m dp 24 where p hm c Qp with Qp being the unit vector in the direction of p . Therefore it follows that by taking a c 4 Q12 - I dmdQpA m 0 m T1 -ớ T2 ị 25 111.7 L J with 0 m T hmN m T being the mean energy of an oscillator and where A m m2 n2c3 plays the role of the density of states. By performing the integral over all the frequencies and orientations in Eq. 25 we finally obtain the expression of the heat interchanged Q12 ơ tỉ -T24 26 where ơ n k4 60ti.3c2 is the Stefan constant. At equilibrium T1 T2 therefore Q12 0 . This expression reveals the existence of a stationary state Saida 2005 of the photon gas emitted at two different temperatures. Note that for a fluid in a temperature gradient the heat current is linear in the temperature difference whereas in our case this linearity is not observed. Despite this fact mesoscopic non-equilibrium thermodynamics is able to derive non-linear laws for the current. In addition if we set T2 0 in Eq. 26 we obtain the heat radiation law of a hot plate at a temperature T1 in vacuum Planck 1959 Q1 TỈ. 27 5. Near-field radiative heat exchange between two NPs In this section we will apply our theory to study the radiative heat exchange between two NPs in the near-field approximation i.e. when the distance d between these NPs satisfies both d ẢT and the near-field condition 2R d 4R with