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