Datenanalyse und Stochastische Modellierung
5. Renewal Processes and Time-Reversal Symmetry

Time reversal symmerty

Often non-reversible dynamics is connected to dissipation

Energy preserving systems are not in general equivalent to time-reversible systems

Thermodynamics

Loschmidt's paradox: Even though the microscopic description of particles is time-reversible, the macroscopic Thermodynamics has no time-reversal symmetry (Entropy decreases)

Detailed Balance

Time reversal symmetry in discrete Markov processes with discrete states \[ x_{i}=A_{ij}x_j \] is called detailed balance: \[ P(x_i)A_{ij} = P(x_j)A_{ji} \]

  • The probability of going from state i to state j is equal to the probability of going from state j to state i.

The Autocorrelation Function

The autocorrelation function is symmetric around t=0 for stationary processes. \[\sum_{t=0}^{T-\Delta} x(t+\Delta) x(t) / \sigma^2 \] is identical to \[\sum_{t=0}^{T-\Delta} x(T-t) x(T-t-\Delta) / \sigma^2 \]

  • Even though the autocorrelation function is time-symmetric, higher-order correlations can still exhibit time reversal symmetry

Gaussian Prozesses

Gaussian Stochastic Processes can be defined by the first and second moment, so stationary Gaussian Processes have time reversal symmetry

  • As an illustration, look at the 4th-order autocorrelation
\[ \dot x(t) = -x(t)/\tau + \xi(t) \] \[ H(t)=\langle x(t)x^3(0)\rangle = e^{-t/\tau} \langle x^4\rangle \] \[ H(-t)=\langle x^3(t)x(0)\rangle = e^{-3t/\tau} \langle x^4\rangle + 3e^{-t/\tau} (1-e^{-2t/\tau}) \langle x^2\rangle^2 \] \[ \Rightarrow H(t)-H(-t) = ( \langle x^4 \rangle - 3\langle x^2 \rangle^2 ) ( e^{-t/\tau} - e^{-3t/\tau} ) \]
  • which is zero in the case of the Gaussian distribution

Generalizing the AR(1) Process

  • Add Poissonian waiting times between the noise increments of the AR(1) process, so the process is no longer Gaussian

Tests for time reversal symmetry

Approximating the data with AR(1)-like dynamics and reconstructing the noise term in forward and backward direction

\[ \mbox{With } \; \tilde\xi_+=x(t)-ax(t-1) \; \mbox{ and } \; \tilde\xi_-=x(t-1)-ax(t) \] \[ \gamma=\frac{\mbox{median}(\tilde\xi_+^2)}{\mbox{median}(\tilde\xi_-^2)}\]

we can derive the type of asymmetry

\[\begin{array}{l} 1 > \gamma \;\;\; \mbox{jump-and-relax}, \\ \gamma = 1 \;\;\; \mbox{time-symmetric}, \\ \gamma > 1 \;\;\; \mbox{build-up-and-reset}. \end{array}\] see Phys. Rev. E. 104,024208

Summary

  • Time reversal symmetry is a property that is not visible in the distribution and the autocorrelation function of the signal
  • Higher order autocorrelations or related measures reveal time reversal symmetry/asymmetry
  • Gaussian processes (and transformations of Gaussian processes) have time reversal symmetry
  • Asymmetric processes can be classified in jump-and-relax and build-up-and-reset dynamics

Back to violations of the premises of the central limit theorem

After discussing processes with not identically distributed random variables in the previous chapter, we now turn to non-independent random variables in the next chapter