Evolutionary games of condensates in coupled birth-death ...denysov/Lectures/Vortrag-NEST.pdf · Evolutionary games of condensates in coupled ... Evolutionary games of condensates

Evolutionary games of condensates in coupledbirth-death processes

Simon Kirschler, Asmar Nayis

Universitat Augsburg

21. March 2016

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

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. Theyoccure 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 inevolutionary 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 states Ei , i = 1, ...,S

each is occupied by Ni ≥ 0 indistinguishable particles

System at time t is given by the occupation number N = (N1,N2, ...,NS)


Introduction

We’re interested in the probability of finding the system in configuration N attime t.The temporal evolution of the probability distribution is given by classical masterequation:

∂tP(N, t) =S∑

i,j=1j 6=i

(Γi←j(Ni − 1,Nj + 1)P(N − ei + ej , t)− Γi←j(Ni ,Nj)P(N, t)

The rate for the transition of particles from Ej to Ei depends linearly on thenumber of particles:

Γi←j = rij(Ni + sij)Nj

with rate constant rij ≥ 0 and constant sij ≥ 0


Introduction

Condensate:- the long-time average of the number of particles of Ei scales linearly with thesystem size

Depleted state:- a state is depleted when the average occupation number scales less than linearlywith the system size.

→ The fraction of particles in a depleted state vanishes in the limit of largesystems and the condesates become macroscopicaly occupied.


Problem illustrationnon-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 rateΓi←j = rij(Ni + 1)Nj with sij = 1 for bosons.The quantum statistics is encoded in the functional form of Γi←j . The rateconstant rij is determined by the microscopic properties of the system and thereservoir.


Problem illustrationstrategy selection in evolutionary game theory

In evolutionary game theory (EGT) the system consists of N interacting agents.Each plays a fixed strategy Ei out of S possible choices E1,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 plays Ej spontaneouslymutate into an agent who plays Ei , one recovers Γi←j = rij(Ni + 1)Nj .

→ Correspondence between incoherently driven-dissipative bosonic systems andstrategy selection in EGT.

→ The states in an incoherently driven-dissipative set-up play an evolutionarygame and the winning states form the condensates.


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 theALVE:

d

dtxi = xi (Ax)i

The matrix A is antisymmetric and encodes the effective transition rates betweenstates (aij = rij − rji ).



The ALVE is solved by,xi (t) = xi (0)et(A〈x〉t)i ,

with the time average of the trajectory 〈x〉t defined as:

〈x〉t =1

t

t∫0

ds x(s).

It can be shown that:

xi (t) ≥ Const(A, x0) > 0 for all t ≥ 0

|(A〈x〉t)i | ≤1

t|log

(xi (t)

xi (0)

)| ≤ Const(A, x0)

tfor all i ∈ I




Production of relative entropy and condensate selection

Theorem:Given an antisymmetric matrix A, it is always possible to find a vector c thatfullfils the following conditions:The entries of c are positive for indices in I ⊆ {1, ...,S} and zero for indices inI = {1, ...,S} − I , whereas the entries of Ac are zero for indices in I and negativefor indices in I .



The global stability properties can be derived by the relative entropy

D(c ‖ x) =∑i∈I

ci log(cixi

)

with the properties of the condensate vector c:

ci > 0 and (Ac)i = 0 for i ∈ I

ci = 0 and (Ac)i < 0 for i ∈ I .

The time derivative of the relative entropy yields:

d

dtD(c ‖ x)(t) = −

S∑i=1

ci∂txixi

= −S∑

i=1

ci (Ax)i =S∑

i=1

(Ac)ixi =∑i∈I

(Ac)ixi

Since (Ac)I < 0 and x > 0 → ∂tD(c ‖ x)(t) < 0.



D(c ‖ x) is bounded because:

0 ≤ D(c ‖ x)(t) = D(c ‖ x)(0) +

t∫0

ds∑i∈I

(Ac)ixi (s) ≤ D(c ‖ x)(0)

→ every concentration xi with i ∈ I remains larger than a positive constant, thatis, xi (t) ≥ Const(A, x0) > 0 for all times t (if xi (t)→ 0 for i ∈ I , it follows thatD →∞, which contradicts the boundedness of D).Furthermore:

0 <

∞∫0

ds xi (s) ≤ D(c ‖ x)(0)

−(Ac)i= Const(A, x0) for every i ∈ I

→ the states with indices in I become depleted as t →∞, that is, xi (t)→ 0 fori ∈ I .





Positive entries of c represent the asymptotic temporal average of condensateconcentrations:

‖〈xi 〉 − c‖∞ ≤Const(A, x0)

t→ 0 as t →∞

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

xi (t) = xi (0)et(A〈x〉t)i

≤ xi (0)et((Ac)i+‖A(〈x〉t−c‖∞)

≤ xi (0)et(Ac)i+Const(A,x0)

= Const(A, x0)et(Ac)i

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



The dynamics of the subsystem do not come to rest. The numbers of particles inthe condensates oscillates

periodic

quasiperiodic

non-periodic


How to find the condensates?

,A = antisymmetric matrix.

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

AI c =

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 ofa random network.


Condensation in large random networks of states


Design of active condensates

Conditions:RPS-condition:

ri−1,i+1 > ri+1,i−1

Attractivity-condition:

3∑j=1

cj rjk >

3∑j=1

cj rkj

RPS cycle


Literature

Evolutionary games of condensates in coupled birht-death processes,Johannes Knebel, Markus F. Weber, Torben Kruger, Erwin Frey , published24. Apr 2015


Documents

Evolutionary games of condensates in coupled birth-death ...denysov/Lectures/Vortrag-NEST.pdf · Evolutionary games of condensates in coupled ... Evolutionary games of condensates