Quantum thermodynamics

Quantum thermodynamics is the study of the relations between two independent physical theories: thermodynamics and quantum mechanics. The two independent theories address the physical phenomena of light and matter.In 1905 Einstein argued that the requirement of consistency between thermodynamics and electromagnetism leads to the conclusion that light is quantized obtaining the relation E = h ν {displaystyle E=h u } . This paper is the dawn of quantum theory. In a few decades quantum theory became established with an independent set of rules. Currently quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. It differs from quantum statistical mechanics in the emphasis on dynamical processes out of equilibrium. In addition there is a quest for the theory to be relevant for a single individual quantum system. Quantum thermodynamics is the study of the relations between two independent physical theories: thermodynamics and quantum mechanics. The two independent theories address the physical phenomena of light and matter.In 1905 Einstein argued that the requirement of consistency between thermodynamics and electromagnetism leads to the conclusion that light is quantized obtaining the relation E = h ν {displaystyle E=h u } . This paper is the dawn of quantum theory. In a few decades quantum theory became established with an independent set of rules. Currently quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. It differs from quantum statistical mechanics in the emphasis on dynamical processes out of equilibrium. In addition there is a quest for the theory to be relevant for a single individual quantum system. There is an intimate connection of quantum thermodynamics with the theory of open quantum systems.Quantum mechanics inserts dynamics into thermodynamics, giving a sound foundation to finite-time-thermodynamics.The main assumption is that the entire world is a large closed system, and therefore, time evolutionis governed by a unitary transformation generated by a global Hamiltonian. For the combined systembath scenario, the global Hamiltonian can be decomposed into: where H S {displaystyle H_{S}} is the system Hamiltonian, H B {displaystyle H_{B}} is the bath Hamiltonian and H S B {displaystyle H_{SB}} is the system-bath interaction.The state of the system is obtained from a partial trace over the combined system and bath: ρ S ( t ) = T r B ( ρ S B ( t ) ) {displaystyle ho _{S}(t)=Tr_{B}( ho _{SB}(t))} .Reduced dynamics is an equivalent description of the system dynamics utilizing only system operators.Assuming Markov property for the dynamics the basic equation of motion for an open quantum system is the Lindblad equation (L-GKS): H S {displaystyle H_{S}} is a (Hermitian) Hamiltonian part and L D {displaystyle L_{D}} : is the dissipative part describing implicitly through system operators V n {displaystyle V_{n}} the influence of the bath on the system.The Markov property imposesthat the system and bath are uncorrelated at all times ρ S B = ρ s ⊗ ρ B {displaystyle ho _{SB}= ho _{s}otimes ho _{B}} .The L-GKS equation is unidirectional and leads any initial state ρ S {displaystyle ho _{S}} to a steady state solution which is an invariant of the equation of motion ρ ˙ S ( t → ∞ ) = 0 {displaystyle {dot { ho }}_{S}(t ightarrow infty )=0} . The Heisenberg picture supplies a direct link to quantum thermodynamic observables. The dynamics of a system observable represented by the operator, O {displaystyle O} , has the form: where the possibility that the operator, O {displaystyle O} is explicitly time-dependent, is included. When O = H S {displaystyle O=H_{S}} the first law of thermodynamics emerges: where power is interpreted as P = ⟨ ∂ H S ∂ t ⟩ {displaystyle P=leftlangle {frac {partial H_{S}}{partial t}} ight angle } and the heat current J = ⟨ L D ∗ ( H S ) ⟩ {displaystyle J=langle L_{D}^{*}(H_{S}) angle } .