Statistical Mechanics and Thermodynamics of Small Systems

Transcription

1 Statistical Mechanics and Thermodynamics of Small Systems Luca Cerino Advisors: A. Puglisi and A. Vulpiani Final Seminar of PhD course in Physics Cycle XXIX Rome, October,

2 Outline of the talk 1. Irreversible Heat Engines 2. Coarse Graining - Modelling of small systems 3. Efficiency at maximum power 0. Small systems Molecular model of a piston (Simulations)

3 Systems far from the Thermodynamic Limit A connection between Statistical Mechanics and Thermodynamics exists in the thermodynamic limit (N ). In systems with few degrees of freedom Fluctuations and average values are of the same order of magnitude a dynamical approach is necessary. Different ensembles are not equivalent (constant energy constant temperature) May thermodynamics be defined for small systems?

4 Why are small systems interesting? Molecular motors inside cells ratchets Granular Systems converting thermal energy into useful work Small Engines. Do they behave like macroscopic engines?

5 Heat Engines: general considerations T o (t) Total time of the cycle Q(t) Ext. Control System E(t) λ(t) Ẇ (t) Desiderata Predict the dependence of the integrated fluxes W and Q on ; Take into account fluctuations (e.g. predict P(W ));

6 A paradigmatic small system A system composed of N O(10 2 ) degrees of freedom ŷ H = N i=1 p 2 i 2m + P2 2M + FY ˆx T o F (+ elastic collision between particles and piston) (+ thermal wall on the left side at temperature T o ) Is it possible to extract mechanical work from this system with a cyclical protocol?

7 Heat Engine: the Ericsson cycle F W F H F L t T o Q 1 Q 2 T H T C t In each segment: W = dt H t = dt F X (t) Q = H W

8 Results of MD simulations [L.C., A. Puglisi and A. Vulpiani, PRE (2015)] D R D E Thermodynamics forces: 0 δ = T H T C T H + T C = F H F L (II) (III) (I) (IV) <W> <Q 1 > <Q 2 > ɛ = F H F L F H + F L = 0.1 T L T H

9 Coarse-graining: can we understand this behavior? Step 1 Identify the relevant (slow-varying) variables of the system. Step 2 Derive a set of coupled Langevin equations for these variables; Step 3 Use stochastic thermodynamics to derive an explicit expression for thermodyn. quantities (W, Q, η...) and associated fluctuations.

10 Model with 3 Macroscopic Variables A coarse grained description is possible in terms of: X piston position V piston velocity T gas kin. energy per particle y = (X X eq (t), V, T T eq (t)) Linear time-dependent Langevin eqn. ẏ = A(t) y + B(t) ξ white noise A is determined via kinetic theory considerations (collisions gas piston and gas thermostat). B is determined a fortiori to restore detailed balance with equilibrium distribution.

11 Comparison with MD simulations <W>/W <Q 1 >/Q 1 <Q 2 >/Q 2 MD Langevin Av. values: Good qualitative agreement (sign inversion, maximum...) Fluctuations: Approx. Gaussian behaviour and same stdev. Discrepancies: Effect of gas inhomogeneities and non-linear effects.

12 An even simpler model... We can pass from 3 variables 2 variables by simply fixing T (t) = T o (t) dx dt = V dv dt = k(t)(x X 2γkB T o (t) 0(t)) γ(t)v + ξ M k(t) = F (t)2 (m + M) M 2 Nk B T o (t) γ(t) = X 0 (t) = (N + 1) k BT o (t) F (t) 2F (t) 2m M πk B T o (t)

13 ...to obtain analytic formulas! With a simpler protocol ( )] 2πt T o (t) = T 0 [1 + δ sin ( )] 2πt F (t) = F 0 [1 + ɛ cos An analytic expression for P(W ) (in the engine regime for small ɛ and δ): Gaussian! 10 =500 MD Ovdmp. 2V 10 MD Ovdmp. 2V P(W/<W >) 1 P(W/<W >) 1 = W/<W > W/<W >

14 Efficiency at maximum power Efficiency: η = W Q in η C = 1 T C T H Carnot efficiency is attained only in the quasi-static limit but when =, the output power vanishes! η = Efficiency at max. power: A general result holds: η < η CA = 1 TC T H......but only maximizing with respect to ɛ = F H F L F H +F L ( fixed)

15 Maximizing the power V ε MM ε Rescaled Output Power V MM ε=0.05 ε=0.10 ε=0.25 ε=0.35 Rescaled Output Power

16 Efficiency at maximum power Rescaled Output Power ε=0.05 ε=0.10 ε=0.25 ε=0.35 η ~ /η CA MD Lin. 2V ε η/η CA

17 Summarizing... ŷ ˆx T o F Rich phenomenology (due to N 1); Fluctuating thermodynamic quantities (due to N ); Non trivial Langevin description (e.g. impossible to define energy from the Lang. Eq.). A good insight into the thermodynamics of small engines!

18 Published Works Cerino, L., Cecconi, F., Cencini, M., and Vulpiani, A. (2016). The role of the number of degrees of freedom and chaos in macroscopic irreversibility. Physica A: Statistical Mechanics and its Applications, 442: Cerino, L., Gradenigo, G., Sarracino, A., Villamaina, D., and Vulpiani, A. (2014). Fluctuations in partitioning systems with few degrees of freedom. Physical Review E, 89(4): Cerino, L. and Puglisi, A. (2015). Entropy production for velocity-dependent macroscopic forces: The problem of dissipation without fluctuations. EPL (Europhysics Letters), 111(4): Cerino, L., Puglisi, A., and Vulpiani, A. (2015). A consistent description of fluctuations requires negative temperatures. Journal of Statistical Mechanics: Theory and Experiment, 2015(12):P Cerino, L., Puglisi, A., and Vulpiani, A. (2015). Kinetic model for the finite-time thermodynamics of small heat engines. Physical Review E, 91(3): Cerino, L., Puglisi, A., and Vulpiani, A. (2016). Linear and nonlinear thermodynamics of a kinetic heat engine with fast transformations. Physical Review E, 93(4):