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Lecture 2 of 4
Quantum Techniques for Stochastic Mechanics, by Jacob Biamonte, QIC 890/891 Selected Advanced Topics in Quantum Information, The University of Waterloo, Waterloo Ontario, Canada, (Spring term 2012).
Given Aug 9th, 2012 in Waterloo Canada
Review of lecture 1
Probabilities vs amplitudes (chapter 4)
Operators (creation and annihilation)
Amoeba field theory
Preamble to stochastic mechanics vs quantum mechanics
Note: the lecture content here is to be read along with the youtube video of the course. It should contain what was written on the board. Without the video, it might not make sense.
A stochastic petri net has
we want to know
one for each $1\leq i \leq k$.
For each transition we end up with a term like this
We will use index free notation.
is the concentration vector and
input vector: $m = (m_1, \ldots, m_k) \in N^k$
output vector: $n = (n_1, \ldots, n_k) \in N^k$
notation: $x^m = x_1^{m_1}\cdots x_k^{m_k}$
then
Example
In birth, one rabbit comes in and two go out. This is a caricature of reality: these bunnies reproduce asexually, splitting in two like amoebas.
In predation, one wolf and one rabbit come in and two wolves go out. This is a caricature of how predators need to eat prey to reproduce.
In death, one wolf comes in and nothing goes out. Note that we are pretending rabbits don’t die unless they’re eaten by wolves.
$R(t)$ rabbits
$W(t)$ wolves
The rate equations of motion are
Let $\psi_{n_1, \ldots, n_k}$ be the probability that we have $n_1$ of the first thing, $n_2$ of the second, etc.
The master equation says how these “things” change with time
It contains all possible “histories” of the possible interactions
Let $n = (n_1, \ldots, n_k) \in N^k$ and $\psi_n = \psi_{n_1}, \ldots, \psi_{n_k}$ and then write a monomial
and express any stochastic state as
A state has
The master equation provides a description of how $\Psi$ changes with time.
$H$ will be built from creation and annihilation operators
(annihilation) $a_i \psi = \frac{d}{d z_i} \psi$
(creation) $a_i^\dagger \psi = z_i \psi$
$r(\tau)$ rate constant for transition $\tau\in T$
$n(\tau)$ output vector
$m(\tau)$ input vector
then
with
$a^{m(\tau)} = a_1^{m_1(\tau)}\cdots a_k^{m_k(\tau)}$ $a^{\dagger m(\tau)} = a_1^{\dagger m_1(\tau)}\cdots a_k^{\dagger m_k(\tau)}$
How can we understand each term?
The first term $a^{\dagger n(\tau)} a^{m(\tau)}$ describe how $m_i(\tau)$ things of the ith species get annihilated and $n_i(\tau)$ things of the ith get created
The second term $a^{\dagger m(\tau)} a^{m(\tau)}$ says how the probability that nothing happens goes down as time passes.
Consider again
here
where
and so
is the probability of having $n_1$ rabbits and $n_2$ wolves. These probabilities evolve according to
order the input and output vectors as $(R, W)$ then
$m(\tau_1) = (1,0)$
$m(\tau_2) = (1,1)$
$m(\tau_3) = (0,1)$
$n(\tau_1) = (2,0)$
$n(\tau_2) = (0,2)$
$n(\tau_3) = (0,0)$
In the lecture, we briefly recalled some of the basic properties of stochastic mechanics, which can be found in the book, as well as several of the blog articles on azimuth.
It shows a world with one state, amoeba with concentration $A(t)$, and two transitions:
• reproduction, where one amoeba turns into two. Let’s call the rate constant for this transition $\alpha$.
• competition, where two amoebas battle for resources and only one survives. Let’s call the rate constant for this transition $\beta$.
Here
which has solutions
• equilibrium. The horizontal blue line corresponds to the case where the initial population $P_0$ exactly equals the carrying capacity. In this case the population is constant.
• dieoff. The three decaying curves above the horizontal blue line correspond to cases where initial population is higher than the carrying capacity. The population dies off over time and approaches the carrying capacity.
• growth. The four increasing curves below the horizontal blue line represent cases where the initial population is lower than the carrying capacity. Now the population grows over time and approaches the carrying capacity.
The deficiency zero theorem gives conditions on the network that allow one to state that the master equation has a unique equilibrium solution.
For a harmonic oscillator,
A coherence state of the harmonic oscillator minimizes $\Delta p \Delta q$ with $\Delta p = \Delta q$
We can write down
For amoebas
now the probability distribution $\psi_n = e^{-c} \frac{c^n}{n!}$ is called a poisson distribution. The expected number of amoebas is
Quantum Techniques for Stochastic Mechanics, John Baez and Jacob Biamonte, (2012).