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47) Field Pulse Transport in Nanoscale 47 where 2 2Vm mv q 2 . 43) is of the form of the Klein – Gordon equation in the potential field V(x, t). For q < 0 Eq. 47) is the modified Klein – Gordon equation. Considering the existence of the attosecond laser with Δt = 1 as = 10-18s, Eq. 47) describes the heat signaling for thermal energy transport induced by ultra-short laser pulses. In the subsequent we will consider the heat transport when V is the Casimir potential. 43), can be repulsive sign (V) = +1 and attractive sign (V) = –1.

20) describes the quantum heat transport for Δt > τ. For heat transport initiated by ultra-short laser pulses, when Δt < τ one obtains the generalized quantum hyperbolic heat transport equation 2T T 2 2V 2 T T. 21) Considering that τ = ħ/mv2, Eq. 21) can be written as follows: 1 2T m T 2Vm 2 T 2T . e 1 2T m T 2Vm 2T T . 23) 42 Janina Marciak-Kozlowska and Miroslaw Kozlowski We seek a solution in the form T ( x, t ) e 2 u( x, t ). 23) After substitution of Eq.

63) we obtain the new form of quantum thermal equation 2 1 2 2 T 2T . e. 2 3 1 2 2 3 3 T 2T . 66) has the form 2 3 2 2 3 3 T 2T . 67) Let us write Eq. M. Ruiz. In one-dimensional case one obtains from Eq. 72) is the diffusion equation. 73) can be rewritten as 2T 3T . 75) of the form T ( x, t ) Ae i ( kx t ) . 76) to Eq. 75) one obtains (ik ) 2 (i ) 3 . e. Eq. 33]. Substituting Eq. 78) to Eq. 79) where λ is mean free path.