Entropy Production and Fluctuation Relations in NonMarkovian Systems

Transcription

1 Entropy Production and Fluctuation Relations in NonMarkovian Systems Tapio Ala-Nissilä Department of Applied Physics and COMP CoE, Aalto University School of Science (formerly Helsinki University of Technology), Espoo, and Department of Physics, Brown University, U.S.A. [A. Kutvonen et al., arxiv: ; J.P. Pekola et al., JSTAT P02033 (2013); J.V. Koski et al., Nat. Phys. 9, 644 (2013)]

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3 Miracles happen at nanoscale! PR Arrow of time PL Suppose the piece is a cube of aluminium, edge length x If x = 5 cm => P R / P L = If x = 5 nm => P R / P L =4.5

4 Stochastic thermodynamics Thermodynamic quantities (heat, energy, entropy) become fluctuating stochastic quantities, which in many cases obey fluctuation relations Jarzynski (1997): Crooks FT (1998): Control parameter dissipative work Seifert (2005): (generally true) etc.

5 Work in Quantum Systems Problem: There is no (unique) Hermitian work operator (there is only the energy operator ) Several possible definitions of quantum work have been proposed M. Esposito, U. Harbola, and S. Mukamel, Rev. Mod. Phys. 81, 1665 (2009) P. Solinas, D. V. Averin, and J. P. Pekola, Phys. Rev. B 87, (2013) S. Suomela, P. Solinas, J.P. Pekola, J. Ankerhold and T. Ala-Nissila, Phys. Rev. B 90, (2014)

6 Work in unitary quantum system Assume unitary time evolution with

7 Work in unitary quantum system The probability to be in state

8 Work in unitary quantum system The probability to be in state The probability to be in state with energy

9 The work done on the system is given by since the system is isolated The Jarzynski average

10 Jarzynski equation is recovered in two-point measurement protocol in the case where the dynamics is unitary Similar to the classical case, if the system is not isolated during the drive, there are problems with the Jarzynski equation...

11 Stochastic forward and reverse paths for a driven (open) system with states and System states Control parameter (drive protocol)

12 Path probabilities with transition rates Forward path Time-reversed path

13 The total entropy change can be written as (Seifert (2005))

14 The total entropy change can be written as (Seifert (2005)) This can be split as

15 The total entropy change can be written as (Seifert (2005)) This can be split as Mathematical identity Conservation of probability

16 The total entropy change can be written as (Seifert (2005)) This can be split as Mathematical identity Conservation of probability Assuming local detailed balance and Boltzmann distribution

17 Using (local) detailed balance & canonical equilibrium: The total entropy expression becomes the Crooks FT The Seifert integral FT becomes the Jarzynski eqn.

18 Single Electron Box (SEB) Electrons are driven one by one from lead to island and stochastically undergo transitions

19 Single Electron Box (SEB) SEB

20 SEB in Equilibrium: Experimental Results Γ + (t ) Γ (t ) + βδu (t) =e f = 4 Hz f = 1 Hz f = 2 Hz

21 Markovian Non-Markovian Nonequilibrium subsystem

22 SEB Model of Overheating by Transitions [J.P. Pekola, A. Kutvonen, T.A-N., JSTAT P02033 (2013)] Heat bath β n Big piece of metal f L (E, β) Transition rates ± Weak coupling Strong coupling Small piece of metal f I ( E, β) Initial equilibrium Γ (t)= GT e 2 f L ( β L, E)[1 f I ( β I, E± ΔU (t))]de Electron-electron relaxation is the fastest timescale in the problem Depends on the energy of the tunneling electron f L (E, β) f I ( E, β+ Δβ ) Tunneling event perturbs the system Non-equilibrium + Γ (t) + βδu (t ) 1 =e [1 Q Δβ+...] 2 Γ (t )

23 Analytic and Numerical Results for Overheated (Non-Markovian) SEB n g (t ) Analytically assume single (double) jump trajectory, expand tunneling rates in Δβ: β+δβ(c,q) 1 β (W ΔF ) e 0 kb 2 =1 β Q 2+ O[( Δβ)2 ] 4C t β (W ΔF ) e Monte Carlo: 1011 repetitions 8000 timesteps Single jump trajectory 95%

24 In overheated SEB the NM entropy flow can be identified!

25 Main New Results:

26 Main New Results: Reversed protocol Requires measurements on environment If there is additional entropy generation, the standard stochastic entropy does not satisfy fluctuation theorems!

27 Summary and Thanks For driven non-equilibrium systems coupling to environment may lead to non-markovian dynamics and violation of FTs Non-Markovian entropy generation can be estimated from reverse stochastic paths Main collaborators in this work: Aki Kutvonen (Aalto) Jukka Pekola (Aalto) Quantum Systems: Samu Suomela, Ivan Savenko, Paolo Solinas, Mikko Möttönen (Juha Salmilehto), Jukka Pekola, Takahiro Sagawa, Joachim Ankerhold, Frank Hekking...