The role of PT phase transition in biology and ecology
Dorje C. Brody
Department of Mathematics
University of Surrey
www.surrey.ac.uk/people/dorje-c-brody

Non-Hermitian Quantum Mechanics 2022: 4 July 2022
-1-The role of PT phase transition in biology and ecology

4 July 2022

-2-

Stochastic unravelling problem
Finding a wave-function-valued random process |ψti such that the corresponding
density matrix ρ̂ = E[|ψtihψt|] obeys the Lindblad equation
1 2
1 2L̂ ρ + ρ̂L̂2(1)∂tρ̂ = −i[Ĥ, ρ̂] + σ L̂ρ̂L̂ −4
2is the stochastic unravelling problem.
A well-known solution is given by the stochastic Schrödinger equation
211 2d|ψti = −iĤ |ψti dt + 2 σ L̂ − hL̂it |ψti dWt − 8 σ L̂ − hL̂it |ψti dt, (2)wherehL̂it = hψt|L̂|ψti,(3)and {Wt} denotes a Brownian motion.

The solution to the stochastic Schrödinger equation (2) can be found byconsidering an underlying signal detection problem.
NH2022c DC Brody 2022

The role of PT phase transition in biology and ecology-3-4 July 2022

A signal-detection problem
Consider the following problem in signal processing.
We have an unknown quantity of interest represented by the random variable L,
that takes the discrete values {li} with the probabilities {pi}.
The true value of L is unknown to the observer, who merely receives information
about the value of the signal L that is obscured by noise.
Assume that noise is additive and is modelled by a Brownian motion {Bt}, andthat the signal is revealed in time at a constant rate σ.
Then the noisy observation of the signal is characterised by the information
process
ξt = σLt + Bt.

NH2022(4)c DC Brody 2022

The role of PT phase transition in biology and ecology

4 July 2022

-4-

The best estimate of L that minimises the quadratic error, given the observed
time series {ξs}s≤t up to time t, is given by the conditional expectation
X
hLit =li P (L = li|{ξs}s≤t) .
(5)
iWriting πit = P(L = li|{ξs}s≤t), we haveP(L = li) ρ(ξt|L = li)Pπit =,P(L=l)ρ(ξ|L=l)jt
jj(6)

where P(L = lj ) = pi and
1(ξ − σlit)2ρ(ξ|L = li) = √exp −.
2t2πt

(7)

Therefore, we deduce thatpi exp σli ξt − 12 σ 2li2t.
πit = P1 2 2j pj exp σlj ξt − 2 σ lj t
NH2022(8)c DC Brody 2022

The role of PT phase transition in biology and ecology

4 July 2022

-5-Taking the stochastic differential of the conditional probability process, we finddπit = σπit li − hLit (dξt − σhLit dt) .
(9)Introducing the process {Wt} according toZ tWt = ξt − σ hLis ds,

(10)

0

we can show that {Wt} thus defined is a standard Brownian motion.

In signal detection, {Wt} is called the innovations process, and it reveals thearrival of new information.
Now if we define φit =√

πit, then we find2
11 2dφit = 2 σ li − hLit φit dWt − 8 σ li − hLit φit dt.

NH2022(11)c DC Brody 2022

The role of PT phase transition in biology and ecology-6-4 July 2022

Signal processing quantum dynamics
With this expression in mind, consider a Hilbert space H associated with aphysical system, on which an operator L̂ is defined, whose eigenvalues are {li}and eigenstates are {|lii}.
Assume that the Hamiltonian Ĥ of the system, whose eigenvalues are {Ei},commutes with L̂.
Let the initial state |ψ0i of the system be pure, and let pi = |hli|ψ0i|2.
Let the process |ψti be given by|ψti =Xi

φit ei(θi−Eit) |lii,(12)where {θi} are arbitrary constant and φit satisfies (11).

NH2022c DC Brody 2022

The role of PT phase transition in biology and ecology-7-4 July 2022

Quantum dynamics for motion tracking
This feature of quantum dynamics resembles, at least at an intuitive level, thebehaviour of biological systems.
With this in mind we attempt to model motions of biological systems – plants inthis case – in response to changing environments using stochastic Schrödingerequations.
To characterise the tracking motion we consider the model:
2
1 2
1
dψt(x) = −iµP̂ ψt(x) dt + 2 σ Q̂ − hQ̂it ψt(x) dWt − 8 σ Q̂ − hQ̂it ψt(x) dt.
(13)
The idea that we propose here therefore is to regard the squared wave functionπt(x) = |ψt(x)|2 as representing the probability distribution of the location of
the light source, as “perceived” by the plant.
Hence the integral
the light source.
NH2022Rx πt(x) dx represents the best estimate of the location ofc DC Brody 2022

The role of PT phase transition in biology and ecology

4 July 2022

-8-

For simplicity let us assume that the initial state is a standard Gaussian state:ψ0(x) = (2π)−1/4 exp(− 14 x2).
The signal detection solution to (13) is then as follows.
We let X be a normal random variable with mean zero and variance one.
Consider the “signal” process Xt = X + µt.
The observation of the signal, however, is obscured by a Brownian noise, giving
rise to the information processZ t
Xs ds + Bt.
(14)
ξt = σ0

Then we find that the full solution to (13) is given by
2 !
1


122(1 + σ t)x − σξt − µt(1 + 2 σ t)
1 + σ 2t 4
exp −ψt(x) =,2π4(1 + σ 2t)where the innovations Brownian motion {Wt} in (13) is related to theNH2022(15)

c DC Brody 2022

The role of PT phase transition in biology and ecology-9-information process {ξt} according toZ tσξs + µs(1 + 12 σ 2s)
Wt = ξt − σds.
21+σ s0
The best estimate for the position of the light source isZ ∞σξt + µt(1 + 12 σ 2t)
x πt(x) dx =.
21+σ t−∞4 July 2022

(16)

(17)

Quantifying the information extracted
We are interested in the question on how much information extraction is needed
so as to deduce the location of the unknown moving target {Xt}.
To quantify the information extraction, we consider the reduction S0 − Sτ of theShannon-Wiener entropy
ZSt = − πt(x) log πt(x)dx, πt(x) = |ψt(x)|2.
(18)
NH2022c DC Brody 2022

The role of PT phase transition in biology and ecology

4 July 2022

- 10 -

Because the entropy process {St} is stochastic, we are interested in theaveraged entropy reduction ∆S = S0 − E[Sτ ] for the information gain.
In the present example, we find that1log 2,(19)2and henace that the amount of information processed to track the motion is at
most of order few bits.
∆S =Information erasure and heat production
It seems reasonable to assume that much of the information processed for plantmovements are not stored in the plant indefinitely.
Information erasure is necessarily accompanied by energy consumption andhence heat production, resulting in an increase of environmental entropy.

NH2022c DC Brody 2022

The role of PT phase transition in biology and ecology- 11 -

4 July 2022

The minimum amount of energy consumption required for the erasure of one bitof information at temperature T is kBT log 2.
Based on our estimate, a daisy flower at summer-night temperature (say, 290 K)
will thus consume at least of order 10−2 eV of energy per each
information-encoding unit.
This in turn results in heat production.
To get a better intuition for the scale involved, consider the circumnutation of acommon bean plant, assuming that each cell contains at most a single such unit.
To identify the location of an object within a ±10◦ angular window with 95%
confidence, it must process between 15 and 20 bits of information.
If the object is removed, this information must be erased.

NH2022c DC Brody 2022

The role of PT phase transition in biology and ecology- 12 -4 July 2022

Raja, et al. Scientific Reports 10, 19465 (2020)
If the plant can process information relatively close to the Landauer limit, say,
104 times the limit, then the cost of erasure is of order 104 eV; about 1% of thetotal energy consumed by the cell.
If information is processed significantly less efficiently, say at the level of oureveryday computers, then the erasure cost becomes too high for survival.

NH2022c DC Brody 2022

The role of PT phase transition in biology and ecology- 13 -4 July 2022

Phase transition in stochastic Schrödinger equation
So far we have considered the case in which either [Ĥ, L̂] = 0 or [Ĥ, L̂] ∝ 1.
In more general situations, the system can exhibit a phase transition, dependingon the overall magnitudes of Ĥ and L̂.
This is the transition associated with the reality of the eigenvalue structures ofthe Liouville operator – like a PT phase transition (Brody & Longstaff, PRR 1,033127, 2019).
In biological or ecological context, L̂ represents adaptation, while Ĥ representschanging environmental conditions.
Thus, to model the survival of species or the breakdown of ecological systems,
manifestly ‘quantum mechanical’ models are needed.

NH2022c DC Brody 2022

The role of PT phase transition in biology and ecology

4 July 2022

- 14 -

Second law in biology
It is tempting to conjecture that biological systems operate by extracting
information from their environments, processing them, and arriving at the best
estimate of the state of the environment for the purpose of adaptation.
This results in the gain of information, and thus reduction in entropy.
However, some, or much of the processed information is lost, resulting inincreasing entropy.
The process of information erasure then must lie at the heart of ageing – orarrow of time – in biological systems.
Work based on:Brody, D.C.
“Open quantum dynamics for plant motions”
Scientific Reports 12, 3042 (2022)
NH2022c DC Brody 2022