### 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