There is great confusion about the Liouville equation and what it represents –or doesn't.

It represents a mechanical system that that undergoes mechanical processes that are strictly reversible, that is, processes that are isentropic.

It does not (in fact, it cannot) represent a thermodynamic system that exchanges heat, heat is not a concept that mechanics understands. It cannot represent the evolution of a system whose entropy changes. And it does not/cannot represent relaxation to equilibrium.

Many have argued, incorrectly, that since the Gibbs entropy, $-\int \rho(\Gamma)\log\rho(\Gamma)) d\Gamma$, remains unchanged in the Liouville equation, this must mean that the Gibbs entropy is wrong, or incomplete, or not exactly the Clausius entropy and so on. This argument blames Liouville's inadequacies on Gibbs.

The Liouville equation is inadequate because it applies deterministic mechanics, and as we know, thermodynamics requires statistical mechanics. The determinism that plagues Liouville arises from the fact that it treats the boundaries of the system as deterministic. As Gibbs put it:

The forces are supposed to be determined for every system by the same law, being functions of the coordinates of the system $q_1,\cdots q_n$, either alone or with the coordinates $a_1$, $a_2$, etc. of certain external bodies. It is not necessary that they should be derivable from a force-function. The external coordinates $a_1$, $a_2$, etc. may vary with the time, but at any given time have fixed values. In this they differ from the internal coordinates $q_1,\cdots q_n$, which at the same time have different values in the different systems considered. (Gibbs, Elementary Principles in Statistical Mechanics, p. 5).

The "external coordinates" represent interactions with the surroundings and are treated as deterministic functions of time. Of course the surroundings are made of particles whose motion depends on degrees of freedom that are not included in the Liouville equation. This is the approximation in the Liouville treatment that robs it of the ability to exhibit thermodynamic behavior.

To return to Jaynes's view, it is our ignorance about the precise state of the surroundings that increases entropy: a real system evolves differently from what Liouville predicts because we are ignorant of information that is necessary to predict the future state. We end up with more microstates than we thought we had at $t=0$, because we did not know the number of microstates available in the surroundings of the system. These microstates "seep" into the system in the form of entropy generation.

There is great confusion about the Liouville equation and what it represents –or doesn't.

It represents a mechanical system that that undergoes mechanical processes that are strictly reversible, that is, processes that are isentropic.

It does not (in fact, it cannot) represent a thermodynamic system that exchanges heat, heat is not a concept that mechanics understands. It cannot represent the evolution of a system whose entropy changes. And it does not/cannot represent relaxation to equilibrium.

Many have argued, incorrectly, that since the Gibbs entropy, $-\int \rho(\Gamma)\log\rho(\Gamma)) d\Gamma$, remains unchanged in the Liouville equation, this must mean that the Gibbs entropy is wrong, or incomplete, or not exactly the Clausius entropy and so on. This argument blames Liouville's inadequacies on Gibbs.

The Liouville equation is inadequate because it applies deterministic mechanics, and as we know, thermodynamics requires statistical mechanics. The determinism that plagues Liouville arises from the fact that it treats the boundaries of the system as deterministic. As Gibbs put it:

The forces are supposed to be determined for every system by the same law, being functions of the coordinates of the system $q_1,\cdots q_n$, either alone or with the coordinates $a_1$, $a_2$, etc. of certain external bodies. It is not necessary that they should be derivable from a force-function. The external coordinates $a_1$, $a_2$, etc. may vary with the time, but at any given time have fixed values. In this they differ from the internal coordinates $q_1,\cdots q_n$, which at the same time have different values in the different systems considered. (Gibbs, Elementary Principles in Statistical Mechanics, p. 5).

The "external coordinates" represent interactions with the surroundings and are treated as deterministic functions of time. Of course the surroundings are made of particles whose motion depends on degrees of freedom that are not included in the Liouville equation. This is the approximation in the Liouville treatment that robs it of the ability to exhibit thermodynamic behavior.

To return to Jaynes's view, it is our ignorance about the precise state of the surroundings that increases entropy: a real system evolves differently from what Liouville predicts because we are ignorant of information that is necessary to predict the future state. We end up with more microstates than we thought we had at $t=0$, because we did not know the number of microstates available in the surroundings of the system. These microstates "seep" into the system in the form of entropy generation.

There is great confusion about the Liouville equation and what it represents –or doesn't.

It represents a mechanical system that that undergoes mechanical processes that are strictly reversible, that is, processes that are isentropic.

It does not (in fact, it cannot) represent a thermodynamic system that exchanges heat, heat is not a concept that mechanics understands. It cannot represent the evolution of a system whose entropy changes. And it does not/cannot represent relaxation to equilibrium.

Many have argued, incorrectly, that since the Gibbs entropy, $-\int \rho(\Gamma)\log\rho(\Gamma)) d\Gamma$, remains unchanged in the Liouville equation, this must mean that the Gibbs entropy is wrong, or incomplete, or not exactly the Clausius entropy and so on. This argument blames Liouville's inadequacies on Gibbs.

The Liouville equation is inadequate because it applies deterministic mechanics, and as we know, thermodynamics requires statistical mechanics. The determinism that plagues Liouville arises from the fact that it treats the boundaries of the system as deterministic. As Gibbs put it:

The forces are supposed to be determined for every system by the same law, being functions of the coordinates of the system $q_1,\cdots q_n$, either alone or with the coordinates $a_1$, $a_2$, etc. of certain external bodies. It is not necessary that they should be derivable from a force-function. The external coordinates $a_1$, $a_2$, etc. may vary with the time, but at any given time have fixed values. In this they differ from the internal coordinates $q_1,\cdots q_n$, which at the same time have different values in the different systems considered. (Gibbs, Elementary Principles in Statistical Mechanics, p. 5).

The "external coordinates" represent interactions with the surroundings and are treated as deterministic functions of time. Of course the surroundings are made of particles whose motion depends on degrees of freedom that are not included in the Liouville equation. This is the approximation in the Liouville treatment that robs it of the ability to exhibit thermodynamic behavior.

To return to Jaynes's view, it is our ignorance about the precise state of the surroundings that increases entropy: a real system evolves differently from what Liouville says because we are ignorant of information that is necessary to predict the future state. We end up with more microstates than we thought we had at $t=0$, because we did not know the number of microstates available in the surroundings of the system. These microstates "seep" into the system in the form of entropy generation.

There is great confusion about the Liouville equation and what it represents –or doesn't.

It represents a mechanical system that that undergoes mechanical processes that are strictly reversible, that is, processes that are isentropic.

It does not (in fact, it cannot) represent a thermodynamic system that exchanges heat, heat is not a concept that mechanics understands. It cannot represent the evolution of a system whose entropy changes. And it does not/cannot represent relaxation to equilibrium.

Many have argued, incorrectly, that since the Gibbs entropy, $-\int \rho(\Gamma)\log\rho(\Gamma)) d\Gamma$, remains unchanged in the Liouville equation, this must mean that the Gibbs entropy is wrong, or incomplete, or not exactly the Clausius entropy and so on. This argument blames Liouville's inadequacies on Gibbs.

The Liouville equation is inadequate because it applies deterministic mechanics, and as we know, thermodynamics requires statistical mechanics. The determinism that plagues Liouville arises from the fact that it treats the boundaries of the system as deterministic. As Gibbs put it:

The forces are supposed to be determined for every system by the same law, being functions of the coordinates of the system $q_1,\cdots q_n$, either alone or with the coordinates $a_1$, $a_2$, etc. of certain external bodies. It is not necessary that they should be derivable from a force-function. The external coordinates $a_1$, $a_2$, etc. may vary with the time, but at any given time have fixed values. In this they differ from the internal coordinates $q_1,\cdots q_n$, which at the same time have different values in the different systems considered. (Gibbs, Elementary Principles in Statistical Mechanics, p. 5).

The "external coordinates" represent interactions with the surroundings and are treated as deterministic functions of time. Of course the surroundings are made of particles whose motion depends on degrees of freedom that are not included in the Liouville equation. This is the approximation in the Liouville treatment that robs it of the ability to exhibit thermodynamic behavior.

To return to Jaynes's view, it is our ignorance about the precise state of the surroundings that increases entropy: a real system evolves differently from what Liouville predicts because we are ignorant of information that is necessary to predict the future state. We end up with more microstates than we thought we had at $t=0$, because we did not know the number of microstates available in the surroundings of the system. These microstates "seep" into the system in the form of entropy generation.