Evolutionary games of condensates in coupled birth-death processes

Transcription

1 Evolutionary games of condensates in coupled birth-death processes Simon Kirschler, Asmar Nayis Universität Augsburg 21. March 2016 Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

2 1 Introduction 2 Problem illustration 3 Antisymmetric Lotka-Voltera equation 4 Production of relative entropy and condensate selection 5 How to find the condensates? 6 Condensation in large random networks of states 7 Design of active condensates Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

3 Introduction Condensation phenomena arise through a collective behaviour of particles. They occure in quantum and classical systems and range over a broad area. We ll look at a driven and dissipative system of bosons and a strategy selection in evolutionary game theorie. How can we derive the states becoming condensates? How does this selection of condensates proceed? Is it possible to construct systems that condense into a specific set of condensates? Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

4 Introduction S non-degenerate states E i, i = 1,..., S each is occupied by N i 0 indistinguishable particles System at time t is given by the occupation number N = (N 1, N 2,..., N S ) Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

5 Introduction We re interested in the probability of finding the system in configuration N at time t. The temporal evolution of the probability distribution is given by classical master equation: t P(N, t) = S (Γ i j (N i 1, N j + 1)P(N e i + e j, t) Γ i j (N i, N j )P(N, t) i,j=1 j i The rate for the transition of particles from E j to E i depends linearly on the number of particles: Γ i j = r ij (N i + s ij )N j with rate constant r ij 0 and constant s ij 0 Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

6 Introduction Condensate: - the long-time average of the number of particles of E i scales linearly with the system size Depleted state: - a state is depleted when the average occupation number scales less than linearly with the system size. The fraction of particles in a depleted state vanishes in the limit of large systems and the condesates become macroscopicaly occupied. Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

7 Problem illustration non-interacting bosons in driven-dissipative systems Conditions: bosonic system that is externally driven by a continuing supply of energy dissipate into the environment exhibit decoherence Such a system can be described by the classical master equation and the rate Γ i j = r ij (N i + 1)N j with s ij = 1 for bosons. The quantum statistics is encoded in the functional form of Γ i j. The rate constant r ij is determined by the microscopic properties of the system and the reservoir. Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

8 Problem illustration strategy selection in evolutionary game theory In evolutionary game theory (EGT) the system consists of N interacting agents. Each plays a fixed strategy E i out of S possible choices E 1, E 2,..., E S. When an agent is defeated he adopts the strategy of his opponent. The rate of change is Γ i j = r ij N i N j. When an agent who plays E j spontaneously mutate into an agent who plays E i, one recovers Γ i j = r ij (N i + 1)N j. Correspondence between incoherently driven-dissipative bosonic systems and strategy selection in EGT. The states in an incoherently driven-dissipative set-up play an evolutionary game and the winning states form the condensates. Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

9 Antisymmetric Lotka-Volterra equation (ALVE) To detect macroscopic occupancies the total number of particles is large (N 1) and the particle density N/S is large. The leading order dynamics of the condensation process can be describes by the ALVE: d dt x i = x i (Ax) i The matrix A is antisymmetric and encodes the effective transition rates between states (a ij = r ij r ji ). Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

10 Antisymmetric Lotka-Volterra equation (ALVE) The ALVE is solved by, x i (t) = x i (0)e t(a x t) i, with the time average of the trajectory x t defined as: It can be shown that: x t = 1 t t 0 ds x(s). x i (t) Const(A, x 0 ) > 0 for all t 0 (A x t ) i 1 ( ) t log xi (t) Const(A, x 0) for all i I x i (0) t Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

11 Antisymmetric Lotka-Volterra equation (ALVE) Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

12 Production of relative entropy and condensate selection Theorem: Given an antisymmetric matrix A, it is always possible to find a vector c that fullfils the following conditions: The entries of c are positive for indices in I {1,..., S} and zero for indices in I = {1,..., S} I, whereas the entries of Ac are zero for indices in I and negative for indices in I. Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

13 Production of relative entropy and condensate selection The global stability properties can be derived by the relative entropy D(c x) = i I c i log( c i x i ) with the properties of the condensate vector c: c i > 0 and (Ac) i = 0 for i I c i = 0 and (Ac) i < 0 for i I. The time derivative of the relative entropy yields: d S dt D(c x)(t) = t x i c i x i i=1 S S = c i (Ax) i = i x i = i=1 i=1(ac) (Ac) i x i i I Since (Ac) I < 0 and x > 0 t D(c x)(t) < 0. Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

14 Production of relative entropy and condensate selection D(c x) is bounded because: t 0 D(c x)(t) = D(c x)(0) + 0 ds i I (Ac) i x i (s) D(c x)(0) every concentration x i with i I remains larger than a positive constant, that is, x i (t) Const(A, x 0 ) > 0 for all times t (if x i (t) 0 for i I, it follows that D, which contradicts the boundedness of D). Furthermore: 0 < 0 ds x i (s) D(c x)(0) (Ac) i = Const(A, x 0 ) for every i I the states with indices in I become depleted as t, that is, x i (t) 0 for i I. Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

15 Production of relative entropy and condensate selection Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

16 Production of relative entropy and condensate selection Positive entries of c represent the asymptotic temporal average of condensate concentrations: x i c Const(A, x 0) t 0 as t The exponentially fast depletion of states with i I can be seen as follows: x i (t) = x i (0)e t(a x t) i x i (0)e t((ac) i + A( x t c ) x i (0)e t(ac) i +Const(A,x 0) = Const(A, x 0 )e t(ac) i Condensate selection occurs exponentially fast at depletion rate (Ac) i Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

17 Production of relative entropy and condensate selection The dynamics of the subsystem do not come to rest. The numbers of particles in the condensates oscillates periodic quasiperiodic non-periodic Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

18 How to find the condensates?,a = antisymmetric matrix. Idea: Remove k-th column and row from A and determine the kernel: 0 A I c =. 0 Then fill the condense vector c with zeros where (Ac) i < 0 (the Ī -case). Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

19 Condensation in large random networks of states Now we look at how the selection of condesates is affected by the connectivity of a random network. Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

20 Condensation in large random networks of states Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

21 Design of active condensates Conditions: RPS-condition: RPS cycle r i 1,i+1 > r i+1,i 1 Attractivity-condition: 3 3 c j r jk > c j r kj j=1 j=1 Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21

22 Literature Evolutionary games of condensates in coupled birht-death processes, Johannes Knebel, Markus F. Weber, Torben Krüger, Erwin Frey, published 24. Apr 2015 Simon Kirschler, Asmar Nayis (Universität Augsburg)Evolutionary games of condensates in coupled birth-death processes 21. March / 21