• The text in italics is copied from that url
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### Introduction

The article starts with the following sentence.
A mathematical or physical process is time-reversible if the dynamics of the process remain well-defined when the sequence of time-states is reversed.
All physical processes are irreversible.
Mathematical equations, which describe these processes, can be time reversible. But that does not mean that the process can evolve backward in time.
A deterministic process is time-reversible if the time-reversed process satisfies the same dynamic equations as the original process; in other words, the equations are invariant or symmetrical under a change in the sign of time.
This is typical a mathematical interpretation but not a physical interpretation.
A typical physical problem is that time has no sign. From a mathematical interpretation time can be negative.
A stochastic process is reversible if the statistical properties of the process are the same as the statistical properties for time-reversed data from the same process.
Time-reversed data from a physical process does not exist.

### 1. Mathematics

In mathematics, a dynamical system is time-reversible if etc.
Dynamical physical system can be considered from a mathematical point of view. The opposite is not true.

### 2. Physics

In physics, the laws of motion of classical mechanics exhibit time reversibility, as long as the operator π reverses the conjugate momenta of all the particles of the system, i.e. p --> -p (T-symmetry).
The laws of motions are description of the evolution of physical process, which include physical parameters. It is important that these parameters are subject of constraints, which have to be followed up.
For example: The physical distance between two object is always greater than 0. Time is always greater than zero. This is a tricky sentence, because exactly what is time.
It is better to claim that the duration between two events, at two different locations, is always greater than zero.
For example: The duration to write/enter the above sentence is positive.
Thermodynamic processes can be reversible or irreversible, depending on the change in entropy during the process.
This sentence is not clear because it requires the definition of entropy.
Note, however, that the fundamental laws that underlie the thermodynamic processes are all time-reversible (classical laws of motion and laws of electrodynamics), which means that on the microscopic level, if one were to keep track of all the particles and all the degrees of freedom, the many-body system processes are all reversible; However, such analysis is beyond the capability of any human being (or artificial intelligence), and the macroscopic properties (like entropy and temperature) of many-body system are only defined from the statistics of the ensembles.
The (fundamental) laws of physics are descriptions of actual performed, experiments. If this is not the case than that should be mentioned.
It
When we talk about such macroscopic properties in thermodynamics, in certain cases, we can see irreversibility in the time evolution of these quantities on a statistical level.
This sentence is too vaque. More detail is required.

### 4. Waves and optics

Time reversal method works based on the linear reciprocity of the wave equation, which states that the time reversed solution of a wave equation is also a solution to the wave equation since standard wave equations only contain even derivatives of the unknown variables.
It should be clearly indicated what time reversal in physics means.
The fact that there are two solutions of the wave equation (One with +t and one with -t) does not mean that mean that the second solution is a valid solution. Both should be demonstrated with actual experiments, including that the results are different.
Thus, the wave equation is symmetrical under time reversal, so the time reversal of any valid solution is also a solution.
Okay.
This means that a wave's path through space is valid when travelled in either direction.
That does not indicate time reversal.

### Reflection 1 - General

Any physical process evolves always forward in time. If not disturbed by any human influence, the process evolves it normal cause. Often this is a state of equilibrium in which the evolution stops. To continue, after reaching equilibrium, something has to be changed. This can be: to add product, to remove product, to change the temperature, to change the pressure etc.
There after two things can happen:
1. The process continues in the same direction. For example: the reaction A + B --> C + D continues to produce the products C and D.
2. The process continues in the opposite direction. For example: the reaction C + D --> A + B starts and produces the products A and B.
What is not the case: time is not reversed

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Created: 20 November 2022

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