Fock Space Techniques for Stochastic Physics. John Baez, Jacob Biamonte, Brendan Fong

Transcription

1 Fock Space Techniques for Stochastic Physics John Baez, Jacob Biamonte, Brendan Fong

2 A Petri net is a way of drawing a finite set S of species, a finite set T of transitions, and maps s, t : T N S saying how many times each species appears in the source (input) and target (output) of each transition: In this famous example from epidemiology: S = {susceptible, resistant, infected} s(infection) = susceptible + infected s(recovery) = infected T = {infection, recovery} t(infection) = 2 infected t(recovery) = resistant

3 Petri nets describe stochastic processes: using an unorthodox analogy between quantum mechanics and probability theory, where probabilities are analogous to amplitudes! Jacob Biamonte and I have a book on this, free online: A Course on Quantum Techniques for Stochastic Mechanics.

4 Suppose we have a system with n possibilities: X = {1,..., n} In quantum theory we consider quantum states: ψ : X C with ψ i 2 = 1 i X In probability theory we consider stochastic states: ψ : X R with ψ i = 1 and ψ i 0 i X

5 An operator U : C n C n that sends quantum states to quantum states is called unitary. An operator U : R n R n that sends stochastic states to stochastic states is called stochastic. Concretely, U is stochastic iff U ij = 1 and U ij 0 i

6 An operator H : C n C n for which exp( ith) is unitary for all t R is called self-adjoint. An operator H : R n R n for which exp(th) is stochastic for all t [0, ) is called infinitesimal stochastic. Concretely, H is infinitesimal stochastic iff H ij = 0 and H ij 0 if i j i

7 If H is self-adjoint, we can describe time evolution of quantum states using Schrödinger s equation: d ψ(t) = ihψ(t) dt If H is infinitesimal stochastic, we can describe time evolultion of stochastic states using the master equation: d ψ(t) = Hψ(t) dt Unitary operators have unitary inverses; stochastic operators rarely have stochastic inverses! So, we only evolve forwards in time in stochastic physics.

8 Suppose we have a Petri net: T s t N S Chemists call an element of N S a complex: it says how many items of each species we have. For example, if then S = {susceptible, resistant, infected} 3 susceptible + 2 resistant + 5 infected N S

9 We call the space of functions ψ : N S R the stochastic Fock space. ψ is a stochastic state if κ N S ψ κ = 1 and ψ κ 0 Here ψ κ is the probability that we have exactly κ i items of the ith species for each i S. If we choose a rate constant r(τ) > 0 for each transition τ T of our Petri net, we can define a Hamiltonian H on the stochastic Fock space. Then the master equation d ψ(t) = Hψ(t) dt describes how stochastic states evolve in time.

10 Let s see how to define the Hamiltonian in this example: Here we have just one species and two transitions: fission: amoeba 2 amoeba competition: 2 amoeba amoeba

11 Suppose ψ n is the probability of having n amoebas. We can summarize this information in a power series: Ψ(z) = ψ n z n n=0 The creation operator a creates an amoeba: a Ψ = zψ The annihilation operator a destroys one: aψ = d dz Ψ We have az n = nz n 1 since there are n amoebas to choose from.

12 We re used to aa a a = 1 for indistinguishable bosons. Can this be right for classical objects if we use probabilities instead of amplitudes? Yes! There s one more way to create an amoeba and then kill one, than to kill one and then create one. But let s try some examples.

13 We would like a Hamiltonian for a process that destroys k amoebas and creates j of them: The obvious guess is a j a k. But this is not infinitesimal stochastic! The right answer has a correction term : a j a k a k a k

14 H = a 1 describes the random creation of amoebas. The master equation has this solution: d Ψ(t) = HΨ(t) dt Ψ(t) = e t(z 1) Ψ(0) If we start with the vacuum state Ψ(0) = 1, where there are no amoebas, at time t we have Ψ(t) = etz e t = e t n t n n! zn so the probability of having n amoebas is e t tn n!. This is just what we expect: a Poisson process.

15 H = a a a describes the random annihilation of amoebas. Using this Hamiltonian, the master equation predicts that the expected number of amoebas decays exponentially. Again, this is just right.

16 The Petri net we care about has two transitions: fission and competition: So, if these have rate constants α and β, we get H = α(a 2 a a a) + β(a a 2 a 2 a 2 )

17 We can express the time evolution operator exp(th) = 1 + th + t2 2! H2 + as a sum over Feynman diagrams:

18 We can also show that in the classical limit where ψ n is very sharply peaked near some very large number, the expected number of amoebas: N(t) = n n ψ n (t) obeys the logistic equation: d N(t) = α N(t) β N(t) 2 dt

19 It might seem hard to find a stationary state for our Hamiltonian HΨ = 0 H = α(a 2 a a a) + β(a a 2 a 2 a 2 ) describing amoeba fission and competition. But we can do it using the Anderson Craciun Kurtz theorem.

20 First, find a stationary solution of the rate equation: In our case this easy: d N(t) = α N(t) β N(t) 2 dt α N(t) β N(t) 2 = 0 N(t) = α/β Then form the coherent state where the expected number of amoebas takes this value: Ψ = e(α/β)z e α/β = 1 e α/β (α/β) n n=0 The Anderson Craciun Kurtz theorem implies HΨ = 0! n! z n

21 All this works quite generally. Suppose we have any Petri net with rate constants: (0, ) r T If there are k different species, write s t N S S = {1,..., k} Write the stochastic Fock space as R[[z 1,..., z k ]], the space of formal power series in k variables. A typical element is Ψ = κ N k ψ κ z κ1 z κ k where ψ κ R.

22 We have annihilation and creation operators on the stochastic Fock space: a i Ψ = Ψ z i a i Ψ = z iψ The Hamiltonian for our Petri net with rate constants is: H = ( r(τ) a t(τ) a s(τ)) a s(τ) τ T where for any complex κ we write a κ = a κ 1 1 aκ k k a κ = a 1κ 1 a κ k k for the operators that annihilate or create that whole complex.

23 In the classical limit, the master equation d Ψ(t) = HΨ(t) dt reduces to the rate equation. This is a nonlinear differential equation describing how the expected number of things of each species: changes with time. N i (t) = κ N S κ i ψ κ (t),

24 The Anderson Craciun Kurtz theorem, together with the deficiency zero theorem, says that: and there exist nonzero equilibrium solutions of the rate equation every such equilibrium solution gives a coherent state Ψ in the stochastic Fock space obeying HΨ = 0 if the Petri net is weakly reversible and has deficiency zero.

25 Anderson, Craciun and Kurtz are chemists. So, instead of Petri nets they use an equivalent formalism, reaction networks. Here we take our Petri net: T and draw a directed graph with: transitions τ T as edges, s t N S complexes in the image of s or t as vertices.

26 For example, this Petri net: corresponds to this reaction network: amoeba competition amoeba + amoeba fission

27 A reaction network is weakly reversible if for any edge (that is, transition): κ τ κ there is a directed path of edges going back: κ τ 0 τ 1 τ n 1 τ κ1 κ n n κ

28 The deficiency of a reaction network is: its number of connected components (as a graph) minus its number of vertices plus the dimension of the subspace spanned by vectors t(τ) s(τ) N S R S where τ ranges over all transitions in T.

29 For example, this reaction network is weakly reversible: amoeba competition 2 amoeba fission It has: 1 connected component, 2 vertices, and the dimension of the subspace spanned by vectors 2 amoeba - amoeba, amoeba - 2 amoeba is 1. Thus its deficiency is = 0, and the theorems apply!

30 These ideas generalize to stochastic field theory, where species have locations, e.g. on a lattice. Stochastic field theory is increasingly important in population biology: From Stacey, Gros and Bar-Yam, Beyond the mean field in host-pathogen spatial ecology, arxiv: