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Clarification: In my original wording of this question, regrettably, I did not make it clear that I am interested in the the Boltzmann/Gibbs/statistical interpretation of it, as opposed to the Carnot/Clausius/"classical" (CCC) interpretation. I know that these two interpretations are supposed to be equivalent, in principle. This equivalence is something I can only take on faith. Be that as it may, for the sake of this question, I would like to exclude the CCC interpretation altogether, and restrict the discussion to the BGSM interperation exclusively. In particular, when I write entropy, I mean Boltzmann's

$$S = k_B \ln W .$$

Textbook discussions of the Second Law of Thermodynamics (SLT) often stress that this law applies only to "closed systems". Or, differently stated: if the system is not closed, its entropy can go down.

The argument is further refined as follows. Suppose that $B$ is a closed system, and that $A$ and $A^{'}$ are "disjoint" subsystems of $B$ such that $B = A + A^{'}$. Then, by SLT,

$$\Delta S_B = \Delta S_A + \Delta S_{A^{'}} > 0\label{a}\tag{1}$$

...but this does not rule out cases where

$$\Delta S_A < 0\label{b}\tag{2}.$$

In such case $\Delta S_{A^{'}}$ would be positive and such that $|\Delta S_{A^{'}}| > |\Delta S_A|$ thereby preserving the inequality in $(\ref{a})$.

My question is: How is this particular aspect of the theory verified experimentally?

IOW, is it possible to set up an actual (as opposed to "Gedanken-") experiment where all three quantities in $(\ref{a})$ are measured, and such that, with the measured values, both $(\ref{a})$ and $(\ref{b})$ hold?

Or is the argument given above simply a purely theoretical (i.e. experimentally unverifiable) construct to explain why apparent reductions in entropy¹ are not necessarily violations of SLT?

^{1 ...such as in the formation of ordered crystals from a disordered collection of molecules in solution.}

Clarification: In my original wording of this question, regrettably, I did not make it clear that I am interested in the the Boltzmann/Gibbs/statistical (BGS) interpretation of it, as opposed to the Carnot/Clausius/"classical" (CCC) interpretation. I know that these two interpretations are supposed to be equivalent, in principle. This equivalence is something I can only take on faith. Be that as it may, for the sake of this question, I would like to exclude the CCC interpretation altogether, and restrict the discussion to the BGS interpretation exclusively. In particular, when I write entropy, I mean Boltzmann's

$$S = k_B \ln W .$$

Textbook discussions of the Second Law of Thermodynamics (SLT) often stress that this law applies only to "closed systems". Or, differently stated: if the system is not closed, its entropy can go down.

The argument is further refined as follows. Suppose that $B$ is a closed system, and that $A$ and $A^{'}$ are "disjoint" subsystems of $B$ such that $B = A + A^{'}$. Then, by SLT,

$$\Delta S_B = \Delta S_A + \Delta S_{A^{'}} > 0\label{a}\tag{1}$$

...but this does not rule out cases where

$$\Delta S_A < 0\label{b}\tag{2}.$$

In such case $\Delta S_{A^{'}}$ would be positive and such that $|\Delta S_{A^{'}}| > |\Delta S_A|$ thereby preserving the inequality in $(\ref{a})$.

My question is: How is this particular aspect of the theory verified experimentally?

IOW, is it possible to set up an actual (as opposed to "Gedanken-") experiment where all three quantities in $(\ref{a})$ are measured, and such that, with the measured values, both $(\ref{a})$ and $(\ref{b})$ hold?

Or is the argument given above simply a purely theoretical (i.e. experimentally unverifiable) construct to explain why apparent reductions in entropy¹ are not necessarily violations of SLT?

^{1 ...such as in the formation of ordered crystals from a disordered collection of molecules in solution.}

Clarification: In my original wording of this question, regrettably, I did not make it clear that I am interested in the the Boltzmann/Gibbs/stat-mech (BGSM) interpretation of it, as opposed to the Carnot/Clausius/classical (CCC) interpretation. I know that these two interpretations are supposed to be equivalent, in principle. This equivalence is something I can only take on faith. Be that as it may, for the sake of this question, though, I would like to exclude the CCC interpretation, and restrict the discussion to the BGSM interperation exclusively. In particular, when I write entropy, I mean Boltzmann's

$$S = k_B \ln W .$$

Textbook discussions of the Second Law of Thermodynamics (SLT) often stress that this law applies only to "closed systems". Or, differently stated: if the system is not closed, its entropy can go down.

The argument is further refined as follows. Suppose that $B$ is a closed system, and that $A$ and $A^{'}$ are "disjoint" subsystems of $B$ such that $B = A + A^{'}$. Then, by SLT,

$$\Delta S_B = \Delta S_A + \Delta S_{A^{'}} > 0\label{a}\tag{1}$$

...but this does not rule out cases where

$$\Delta S_A < 0\label{b}\tag{2}.$$

In such case $\Delta S_{A^{'}}$ would be positive and such that $|\Delta S_{A^{'}}| > |\Delta S_A|$ thereby preserving the inequality in $(\ref{a})$.

My question is: How is this particular aspect of the theory verified experimentally?

IOW, is it possible to set up an actual (as opposed to "Gedanken-") experiment where all three quantities in $(\ref{a})$ are measured, and such that, with the measured values, both $(\ref{a})$ and $(\ref{b})$ hold?

Or is the argument given above simply a purely theoretical (i.e. experimentally unverifiable) construct to explain why apparent reductions in entropy¹ are not necessarily violations of SLT?

^{1 ...such as in the formation of ordered crystals from a disordered collection of molecules in solution.}

Clarification: In my original wording of this question, regrettably, I did not make it clear that I am interested in the the Boltzmann/Gibbs/statistical interpretation of it, as opposed to the Carnot/Clausius/"classical" (CCC) interpretation. I know that these two interpretations are supposed to be equivalent, in principle. This equivalence is something I can only take on faith. Be that as it may, for the sake of this question, I would like to exclude the CCC interpretation altogether, and restrict the discussion to the BGSM interperation exclusively. In particular, when I write entropy, I mean Boltzmann's

$$S = k_B \ln W .$$

Textbook discussions of the Second Law of Thermodynamics (SLT) often stress that this law applies only to "closed systems". Or, differently stated: if the system is not closed, its entropy can go down.

The argument is further refined as follows. Suppose that $B$ is a closed system, and that $A$ and $A^{'}$ are "disjoint" subsystems of $B$ such that $B = A + A^{'}$. Then, by SLT,

$$\Delta S_B = \Delta S_A + \Delta S_{A^{'}} > 0\label{a}\tag{1}$$

...but this does not rule out cases where

$$\Delta S_A < 0\label{b}\tag{2}.$$

In such case $\Delta S_{A^{'}}$ would be positive and such that $|\Delta S_{A^{'}}| > |\Delta S_A|$ thereby preserving the inequality in $(\ref{a})$.

My question is: How is this particular aspect of the theory verified experimentally?

IOW, is it possible to set up an actual (as opposed to "Gedanken-") experiment where all three quantities in $(\ref{a})$ are measured, and such that, with the measured values, both $(\ref{a})$ and $(\ref{b})$ hold?

Or is the argument given above simply a purely theoretical (i.e. experimentally unverifiable) construct to explain why apparent reductions in entropy¹ are not necessarily violations of SLT?

^{1 ...such as in the formation of ordered crystals from a disordered collection of molecules in solution.}

Textbook discussions of the Second Law of Thermodynamics (SLT) often stress that this law applies only to "closed systems". Or, differently stated: if the system is not closed, its entropy can go down.

The argument is further refined as follows. Suppose that $B$ is a closed system, and that $A$ and $A^{'}$ are "disjoint" subsystems of $B$ such that $B = A + A^{'}$. Then, by SLT,

$$\Delta S_B = \Delta S_A + \Delta S_{A^{'}} > 0\label{a}\tag{1}$$

...but this does not rule out cases where

$$\Delta S_A < 0\label{b}\tag{2}.$$

In such case $\Delta S_{A^{'}}$ would be positive and such that $|\Delta S_{A^{'}}| > |\Delta S_A|$ thereby preserving the inequality in $(\ref{a})$.

My question is: How is this particular aspect of the theory verified experimentally?

IOW, is it possible to set up an actual (as opposed to "Gedanken-") experiment where all three quantities in $(\ref{a})$ are measured, and such that, with the measured values, both $(\ref{a})$ and $(\ref{b})$ hold?

Or is the argument given above simply a purely theoretical (i.e. experimentally unverifiable) construct to explain why apparent reductions in entropy¹ are not necessarily violations of SLT?

^{1 ...such as in the formation of ordered crystals from a disordered collection of molecules in solution.}

Clarification: In my original wording of this question, regrettably, I did not make it clear that I am interested in the the Boltzmann/Gibbs/stat-mech (BGSM) interpretation of it, as opposed to the Carnot/Clausius/classical (CCC) interpretation. I know that these two interpretations are supposed to be equivalent, in principle. This equivalence is something I can only take on faith. Be that as it may, for the sake of this question, though, I would like to exclude the CCC interpretation, and restrict the discussion to the BGSM interperation exclusively. In particular, when I write entropy, I mean Boltzmann's

$$S = k_B \ln W .$$

Textbook discussions of the Second Law of Thermodynamics (SLT) often stress that this law applies only to "closed systems". Or, differently stated: if the system is not closed, its entropy can go down.

The argument is further refined as follows. Suppose that $B$ is a closed system, and that $A$ and $A^{'}$ are "disjoint" subsystems of $B$ such that $B = A + A^{'}$. Then, by SLT,

$$\Delta S_B = \Delta S_A + \Delta S_{A^{'}} > 0\label{a}\tag{1}$$

...but this does not rule out cases where

$$\Delta S_A < 0\label{b}\tag{2}.$$

In such case $\Delta S_{A^{'}}$ would be positive and such that $|\Delta S_{A^{'}}| > |\Delta S_A|$ thereby preserving the inequality in $(\ref{a})$.

My question is: How is this particular aspect of the theory verified experimentally?

IOW, is it possible to set up an actual (as opposed to "Gedanken-") experiment where all three quantities in $(\ref{a})$ are measured, and such that, with the measured values, both $(\ref{a})$ and $(\ref{b})$ hold?

Or is the argument given above simply a purely theoretical (i.e. experimentally unverifiable) construct to explain why apparent reductions in entropy¹ are not necessarily violations of SLT?

^{1 ...such as in the formation of ordered crystals from a disordered collection of molecules in solution.}