Trying to reconcile two definitions of Entropy

[badatstuff]

My question is regarding a few descriptions of Entropy. I'm actually unsure if my understanding of each version of entropy is correct, so I'm looking for a two birds in one stone answer of fixing my misunderstanding of each and then hopefully linking them together.

1) A measure of the tendency to disperse heat: eg maxwells relations (dS/sU)_[v constant] = 1/T says if you increase the energy of a system while T is low, it has more of an affect on entropy than if you increased the energy by the same amount on a system with high T.

If you place two systems S_1 and S_2 in contact with each other with temperatures T_1 > T_2 then eventually it reaches equilibrium. But the 1/T relation means the entropy of the colder body increases faster than loss of entropy of the warmer body?

2) Entropy is a measure of "disorder" eg a typical examples given in textbooks - someone unbiased of the final outcome stacking of books on a book shelf in completely random orientations, the odds of them being all in alphabetically in order is low due to the number of states. Entropy is a measure of the number of states. You can reduce the number of states by giving the book stacker more information, telling him he can stack them upright reducing the number of permutations and reducing entropy. In terms of system of particles more energy is more accessible states and so higher entropy?

I'm struggling to see the similarities between having a lack of information and seeing a colder body and wanting give it heat/feel the need to ruin its "order" too and make it disordered.

Thanks for your time

 

[DrClaude]

1) A measure of the tendency to disperse heat: eg maxwells relations (dS/sU)_[v constant] = 1/T says if you increase the energy of a system while T is low, it has more of an affect on entropy than if you increased the energy by the same amount on a system with high T.

Thats equation says actually more about temperature than it does about entropy. You can take it as the thermodynamic definition of temperature.

If you place two systems S_1 and S_2 in contact with each other with temperatures T_1 > T_2 then eventually it reaches equilibrium. But the 1/T relation means the entropy of the colder body increases faster than loss of entropy of the warmer body?

Correct, so the overall entropy increases when heat flows from warm to cold.

2) Entropy is a measure of "disorder" eg a typical examples given in textbooks - someone unbiased of the final outcome stacking of books on a book shelf in completely random orientations, the odds of them being all in alphabetically in order is low due to the number of states. Entropy is a measure of the number of states. You can reduce the number of states by giving the book stacker more information, telling him he can stack them upright reducing the number of permutations and reducing entropy. In terms of system of particles more energy is more accessible states and so higher entropy?

I'm struggling to see the similarities between having a lack of information and seeing a colder body and wanting give it heat/feel the need to ruin its "order" too and make it disordered.

You have to be careful with the "order" analogy. It breaks down very rapidly (especially since it can be hard to define what is ordered).

Do you know the difference between a macrostate and a microstate?

 

[badatstuff]

Thank you for your reply.

Do you know the difference between a macrostate and a microstate?

microstates are the different ways you can rearrange your system while remaining in the same macrocscopic state, would that be a correct difference?

You have to be careful with the "order" analogy. It breaks down very rapidly (especially since it can be hard to define what is ordered).

Throughout my undergraduate studies senior lectures kept referring to disorder/order, and then when I went online to clarify what it meant people didn't seem to like that definition so I left "disorder" in quotations.

I'm currently doing something on the Ising model, and this was a rabbit hole I entered. I'm trying to solidify my understanding so I can look at what entropy means for the Ising model eg the curie temperature and the break down of magnetism, the relationship between the "disorder" of the spins at high temperature and entropy as well as its involvement in phase transitions (if any)

 

[NFuller]

I'm struggling to see the similarities between having a lack of information and seeing a colder body and wanting give it heat/feel the need to ruin its "order" too and make it disordered.

Macrostates with a greater number of accessible microstates are higher entropy states than macrostates with fewer microstates. Thus the number of high entropy configurations is larger than the number of low entropy configurations. Since the particle dynamics are considered stochastic, these states are selected at random. This means that the system will have a preference for the high entropy states simply because there are more of them.

If the system is allowed to evolve in time from a non-equilibrium state to an equilibrium state, the system will naturally keep moving between macrostates at random until it finds the one with an overwhelmingly large number of microstates. At this point the system will keep randomly selecting this same configuration and will have reached thermal equilibrium. The second law says that this equilibrium state is also a maximum entropy state. Thus states with a large number of microstates correspond to high entropy states. I recently wrote an article detailing this here https://www.physicsforums.com/insights/statistical-mechanics-part-equilibrium-systems/

 

[badatstuff]

Macrostates with a greater number of accessible microstates are higher entropy states than macrostates with fewer microstates. Thus the number of high entropy configurations is larger than the number of low entropy configurations. Since the particle dynamics are considered stochastic, these states are selected at random. This means that the system will have a preference for the high entropy states simply because there are more of them.

If the system is allowed to evolve in time from a non-equilibrium state to an equilibrium state, the system will naturally keep moving between macrostates at random until it finds the one with an overwhelmingly large number of microstates. At this point the system will keep randomly selecting this same configuration and will have reached thermal equilibrium. The second law says that this equilibrium state is also a maximum entropy state. Thus states with a large number of microstates correspond to high entropy states. I recently wrote an article detailing this here https://www.physicsforums.com/insights/statistical-mechanics-part-equilibrium-systems/

Thanks very much for your explanation, as well as the article. Cleared a lot up for me!