Evolutionary games of condensates in coupled birth-death processes

Evolutionary games of condensates in coupled birth-death processes

Simon Kirschler, Asmar Nayis

Universit¨at Augsburg

21. March 2016

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

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?

Introduction

S non-degenerate statesE_i,i= 1, ...,S

each is occupied byNi≥0 indistinguishable particles

System at timet is given by the occupation numberN= (N1,N2, ...,NS)

Introduction

We’re interested in the probability of finding the system in configurationN at timet.

The temporal evolution of the probability distribution is given by classical master equation:

∂_tP(N,t) =

S

X

i,j=1 j6=i

(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)

The rate for the transition of particles fromE_j toE_i depends linearly on the number of particles:

Γ_i←j =rij(Ni+sij)Nj

with rate constantrij ≥0 and constantsij≥0

Introduction

Condensate:

- the long-time average of the number of particles ofEi 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.

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=rij(Ni+ 1)Nj withsij= 1 for bosons.

The quantum statistics is encoded in the functional form of Γ_i←j. The rate constantrij is determined by the microscopic properties of the system and the reservoir.

Problem illustration

strategy selection in evolutionary game theory

In evolutionary game theory (EGT) the system consists of N interacting agents.

Each plays a fixed strategyEi out ofS possible choicesE1,E2, ...,ES. When an agent is defeated he adopts the strategy of his opponent.

The rate of change is Γi←j =rijNiNj. When an agent who playsEj spontaneously mutate into an agent who playsEi, one recovers Γi←j =rij(Ni+ 1)Nj.

→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.

Antisymmetric Lotka-Volterra equation (ALVE)

To detect macroscopic occupancies the total number of particles is large (N1) and the particle densityN/S is large.

The leading order dynamics of the condensation process can be describes by the ALVE:

d

dtx_i =x_i(Ax)_i

The matrix A is antisymmetric and encodes the effective transition rates between states (a_ij=r_ij−r_ji).

Antisymmetric Lotka-Volterra equation (ALVE)

The ALVE is solved by,

x_i(t) =x_i(0)e^t(Ahxit)i, with the time average of the trajectoryhxit defined as:

hxit= 1 t

t

Z

0

ds x(s).

It can be shown that:

xi(t)≥Const(A,x0)>0for all t ≥0

|(Ahxit)i| ≤ 1 t|log

xi(t) xi(0)

| ≤ Const(A,x0)

t for all i∈I

Antisymmetric Lotka-Volterra equation (ALVE)

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 inI ⊆ {1, ...,S}and zero for indices in I ={1, ...,S} −I, whereas the entries ofAc are zero for indices inI and negative for indices inI.

Production of relative entropy and condensate selection

The global stability properties can be derived by the relative entropy D(ckx) =X

i∈I

c_ilog(c_i xi

) with the properties of the condensate vector c:

c_i>0and (Ac)_i= 0 for i∈I ci = 0and (Ac)i<0 for i∈I. The time derivative of the relative entropy yields:

d

dtD(ckx)(t) =−

S

X

i=1

ci

∂txi

xi

=−

S

X

i=1

ci(Ax)i =

S

X

i=1

(Ac)ixi=X

i∈I

(Ac)ixi

Production of relative entropy and condensate selection

D(ckx) is bounded because:

0≤D(ckx)(t) =D(ckx)(0) +

t

Z

0

dsX

i∈I

(Ac)ixi(s)≤D(ckx)(0)

→every concentrationx_i withi∈I remains larger than a positive constant, that is,xi(t)≥Const(A,x0)>0 for all timest (ifxi(t)→0 fori∈I, it follows that D→ ∞, which contradicts the boundedness ofD).

Furthermore:

0<

∞

Z

0

ds xi(s)≤ D(ckx)(0)

−(Ac)i

=Const(A,x0) for every i∈I

→the states with indices inI become depleted ast→ ∞, that is,x_i(t)→0 for

Production of relative entropy and condensate selection

Production of relative entropy and condensate selection

Positive entries ofc represent the asymptotic temporal average of condensate concentrations:

khxii −ck∞≤ Const(A,x₀)

t →0 as t → ∞

The exponentially fast depletion of states withi∈I can be seen as follows:

xi(t) =xi(0)e^t(Ahxit)i

≤x_i(0)e^{t((Ac)i+kA(hxit−ck∞)}

≤x_i(0)e^{t(Ac)i+Const(A,x0)}

=Const(A,x₀)e^t(Ac)i

→Condensate selection occurs exponentially fast at depletion rate|(Ac)i|

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

How to find the condensates?

,A = antisymmetric matrix.

Idea:

Remove k-th column and row from A and determine the kernel:

A^Ic=





 0

... 0







Then fill the condense vector c with zeros where (Ac)_i<0 (the ¯I-case).

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.

Condensation in large random networks of states

Design of active condensates

Conditions:

RPS-condition:

r_i−1,i+1>r_i+1,i−1 Attractivity-condition:

3

X

j=1

c_jr_jk >

3

X

j=1

c_jr_kj

RPS cycle

Literature

Evolutionary games of condensates in coupled birht-death processes, Johannes Knebel, Markus F. Weber, Torben Kr¨uger, Erwin Frey , published 24. Apr 2015