22.4: Analyzing Trajectories

Analyzing Trajectories

Waiting‐Time Distributions, P

τ_W: Waiting time between arriving and leaving a state or P(k,t)

P_k: Probability of making k jumps during a time interval, t. → Survival probability

P_w: Probability of waiting a time τ_w between jumps? Waiting time distribution → FPT distribution

Let’s relate these...

Assume independent events. No memory of history – where it was in trajectory.

Flux: \(\dfrac{dP_R}{dt}=J\)

J: Probability of jump during ∆t. ∆t is small enough that J\(\ll\) 1, but long enough to lose memory of earlier configurations.

The probability of seeing k jumps during a time interval t, where t is divided into N intervals of width Δt (t = N∆t) is given by the binomial distribution

\[P(k,N)=\dfrac{N!}{k!(N-k)!}J^k(1-J)^{N-k} \]

Here N≫k. Define rate λ in terms of the average number of jumps per unit time

\[ \lambda = \dfrac{\langle k \rangle}{t}=\dfrac{1}{\langle \tau_W \rangle} \nonumber \]

\[J=\lambda \Delta t \rightarrow J=\dfrac{\lambda t}{N} \nonumber \]

Substituting this into eq. (22.4.1) Error! Reference source not found.. For N ≫ k, recognize

\[ (1-J)^{N-k} \approx (1-J)^N = \left( 1-\dfrac{\lambda t}{N} \right)^N \approx e^{-\lambda t} \nonumber \]

The last step is exact for lim N → ∞.

Poisson distribution for the number of jumps in time t.

\[ \langle P(k,t) \rangle = \langle \lambda t \rangle = \dfrac{\lambda t}{\langle P^2(k,t)\rangle^{1/2}}=(\lambda t)^{1/2} \nonumber \]

Fluctuations: \( \sigma / \langle P(k,t) \rangle = (\lambda t)^{-1/2} \)

OK, now what about P_w the waiting time distribution?

Consider the probability of not jumping during time t:

\[P_k(0,t) = e^{-\lambda t} \nonumber \]

As you wait longer and longer, the probability that you stay in the initial state drops exponentially. Note that P_k(0, t) is related to P_w by integration over distribution of waiting times.

\[ \int_{t}^{\infty}P_w(t')dt'=P(0,t)=e^{-\lambda t} \nonumber \]

\[ \int_{t}^{\infty}P_wdt \rightarrow \text {probability of staying for t} \nonumber \]

\[ \int_{0}^{t}P_wdt \rightarrow \text {probability of jumping within t} \nonumber \]

Probability of jumping between t and t+∆t:

Probability of no decay for time <t decay on last

\[\begin{aligned}
P_{w}(t) \Delta t &=\overbrace{\left(1-\langle k\rangle \Delta t_{1}\right)} \overbrace{\left(1-\langle k\rangle \Delta t_{2}\right) \ldots\left(1-\langle k\rangle \Delta t_{N}\right)} \overbrace{k \Delta t} \\ &=(1-\langle k\rangle \Delta t)^{N} k \Delta t \approx k e^{-k t} \Delta t \end{aligned} \nonumber \]

\[\begin{aligned} P_{w} &=\lambda e^{-\lambda t} \\ \langle\tau\rangle &=\int_{0}^{\infty} t p_{w}(t) t
\end{aligned} \nonumber \]

\[ \begin{aligned} \langle\tau_{w}\rangle &=1 / \lambda \rightarrow \text {the average waiting time is the lifetime} (1/\lambda)\\
\langle\tau_{w}^{2}\rangle-\langle\tau_{w}\rangle^{2} &= (1 / \lambda)^{2} \end{aligned} \nonumber \]

Reduction of Complex Kinetics from Trajectories

• Integrating over trajectories gives probability densities.
• Need to choose a region of space to integrate over and thereby define states:

• States: Clustered regions of phase space that have high probability or long persistence.
• Markovian states: Spend enough time to forget where you came from.
• Master equation: Coupled first order differential equations for the flow of amplitude between states written in terms of probabilities.

\[ \dfrac{dP_m}{dt}=\sum_{n}k_{n\rightarrow m}P_n-\sum_{n}k_{m \rightarrow n}P_m \nonumber \]

\(k_{n\rightarrow m}\) is rate constant for transition from state n to state m. Units: probability/time. Or in matrix form: P=kP where k is the transition rate matrix. With detailed balance, conservation of population all initial conditions will converge on equilibrium