Twirl: Equating Quantum and Thermodynamic Entropy Productions október 6. 1 / 10

Transcription

1 Twirl: Equating Quantum and Thermodynamic Entropy Productions Lajos Diósi Wigner Center, Budapest október 6. Acknowledgements go to: Hungarian Scientific Research Fund under Grant No EU COST Action MP1006 Fundamental Problems in Quantum Physics Twirl: Equating Quantum and Thermodynamic Entropy Productions október 6. 1 / 10

2 We search for a graceful non-unitary map W whose von Neumann entropy gain coincides with the calculated thermodynamic entropy production. Our candidate for W is the so-called twirl introduced to quantum information theory by Bennett, also used in the theory of quantum reference frames. Relevance of frame averaging (twirling) for real world irreversibility is outlined. Twirl: Equating Quantum and Thermodynamic Entropy Productions október 6. 2 / 10

3 1 How to equate S Qu with S Th? 2 Simplest example: mechanical friction 3 Graceful irreversible map in MB gas 4 Dissipation in abstract quantum gas 5 Dissipating the external work 6 Twirl W 7 Outlook: Nature s forgetfulness Twirl: Equating Quantum and Thermodynamic Entropy Productions október 6. 3 / 10

4 How to equate S Qu with S Th? How to equate S Qu with S Th? In thermal equilibrium: S Qu = S Th. In non-equilibrium: conflict between macro- and microdynamics. Second law of thermodynamics: S Th 0 Microscopic reversibility: S Qu = 0 Resolution: coarse grained microscopic dynamics! Questions remain: Answers: How much coarse graining? What coarse graining? As much as to equate S Qu with S Th Find graceful irreversible map, universal enough Twirl: Equating Quantum and Thermodynamic Entropy Productions október 6. 4 / 10

5 Simplest example: mechanical friction Simplest example: mechanical friction Irreversible process, double request: Calculable S Th Transparent microscopic dynamics Choice: mechanical friction ds Th /dt = (η V ) V /k B T = ηβv 2 Enjoy simplicity of Maxwell-Boltzmann gas: p 2mV p V Dragging the disk at velocity V needs force 2 νmv. ν= collision rate η=2νm= friction const. ds /dt= 2νmβV 2 Th ds /dt= 0 since collisions are reversible MB Twirl: Equating Quantum and Thermodynamic Entropy Productions október 6. 5 / 10

6 Graceful irreversible map in MB gas Graceful irreversible map in MB gas We need 2mβV 2 MB-entropy production per collision! S MB = S(ρ) = ρ log ρdp 1 dp 2... dp N (k B = 1) ( ) Pre-collision microscopic state: ρ β (p 1, p 2,..., p r,..., p N ) = N exp β N pr 2 2m Post-collision microscopic state (the r th molecule collided): ρ r β(p 1, p 2,..., p r,..., p N ) = ρ β (p 1, p 2,..., p r 2mV,..., p N ) Zero entropy production per collision: S MB = S(ρ r β ) S(ρ β) = 0 Coarse-graining W of ρ r β : erase collider s identity r! ρ r β Wρ r β = 1 N ρ r β N D. 2002: S(Wρ r β ) S(ρr β ) = 2mβV 2 (in leading order of V ) r=1 r=1 Twirl: Equating Quantum and Thermodynamic Entropy Productions október 6. 6 / 10

7 Dissipation in abstract quantum gas Dissipation in abstract qubit gas Initial state: ˆρ β = ˆρ N with N equilibrium qubits ˆρ = N e βĥ Random unitary collisions on external disk : ˆρ ˆσ = Û ˆρÛ ˆρ β = ˆρ N ˆρ r β = ˆρ (r 1) ˆσ ˆρ (N r) External work is tr(ĥ ˆσ) tr(ĥ ˆρ), postulate it has gone dissipated. S Th = β[tr(ĥ ˆσ) tr(ĥ ˆρ)] = S(ˆσ ˆρ) 0 Collision is reversible, yields S Qu = 0. Impose irreversible map W, re-symmetrize the post-collision state: S Qu = S(W ˆρ r β) S(ˆρ β ) D. et al 2007 To equate S Qu with S Th, the following must be true for N : S(W ˆσ ˆρ (N 1) ) S(ˆσ ˆρ (N 1) ) S(ˆσ ˆρ) Proof: Csiszár et al Twirl: Equating Quantum and Thermodynamic Entropy Productions október 6. 7 / 10

8 Dissipating the external work Dissipating the external work Homogeneous equilibrium state: ρ =N e βh β External field perturbs locally: ρ > ρ = β β U ρβ U Postulate the work is dissipated to heat, then: Still: S Qu = S( ρ β ) S( ρ β ) = 0 What irreversible map W can equate S Qu with S Th? Make ρ β forget location of perturbation: W ρ β = lim 1_ e ixp ρ e ixp dx V OO V β V To equate S Qu with S Th, this must hold: S( W ρ β ) S( ρ β ) = S( ρ β ρ β ) Proof: D S Th = S( ρ β ρ β ) > 0 Twirl: Equating Quantum and Thermodynamic Entropy Productions október 6. 8 / 10

9 Twirl W Twirl W is a non-unitary (irreversible) map (Bennett et al. 1996): ˆρ W ˆρ Û(g)ˆρÛ (g)dg Û(g): unitary representation of a group; dg: Haar measure Friction in ideal gas: permutation group of molecules/qubits Dissipation of local work: translational group Û(g)ĤÛ (g)=ĥ: no harm to dynamics, graceful entropy production Postulating S Qu = S Th yields non-trivial conjecture: lim V [ S(WÛ ˆρ βû ) S(ˆρ β ) Fenomenology led us to exact math. ] = lim V S(Û ˆρ βû ˆρ β ), Twirl: Equating Quantum and Thermodynamic Entropy Productions október 6. 9 / 10

10 Outlook: Nature s forgetfulness Outlook: Nature s forgetfulness What is the mechanism of microscopic irreversibility? Is there a graceful way Nature is forgetting microscopic data? Is there a universal non-unitary mechanism? Can it produce as much entropy as expected thermodynamically? Answer: 3xYES If Nature is twirling (shaking?) reference frames, She is producing just the observed thermodynamic entropy. Whether this is the real and ultimate way for Nature to forget microscopic data remains an open question. Twirl: Equating Quantum and Thermodynamic Entropy Productions október / 10