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5

Entropy is subjective in the sense that you get to pick which macroscopic observables you care about keeping track of (usually, for instance, you care about things like temperature, pressure, etc.). Once you've defined the macroscopic observables, entropy is defined as the logarithm of the number of possible microstates that give rise to those macroscopic ...


4

You're missing a minus in the entropy definition - $S=-Tr(\rho\ln\rho)$ Entropy of a unitarily evolving system (doesn't matter in which picture) is conserved (The entropy is a trace of a function of the density matrix "operator" thus it depend solely on the eigenvalues of it's input operator, but the eigenvalues of the density matrix don't change under ...


4

In macroscopic units it should be $$S=-R\alpha \log(\alpha e^{-S_1/R})-R(1-\alpha)\log\Big(1-\alpha)e^{-S_2/R}\Big) \\=\alpha \Big(S_1-R\log\alpha\Big)+(1-\alpha)\Big(S_2-R\log(1-\alpha)\Big),$$ where $R$ is the universal gas constant. In the pure case, this reduces to the textbook formula. But such a formula cannot be true in general. The general formula ...


3

The entropy increases always. There is nothing unusual about the example of planet formation. Either one of two things has to happen when the dust contracts into a planet: The planet heats up as the dust contracts, increasing entropy. Heat is radiated into empty space, decreasing the entropy of the collection of dust but increasing the entropy of the rest ...


2

Antimatter increase in entropy over time. We can verify this with a thought experiment. Take ten positrons. Put five in one side of a chamber with a barrier and then the other 5 on the other side of the barrier in the same chamber. The chamber and barrier are also made of antimatter. The positrons repel each other and so each have a certain amount of kinetic ...


2

Suppose you start with a system in some state $P_1, V_1, T_1$ and you add some quantity of heat $\Delta Q$ to it so the system changes to a different state $P_2, V_2, T_2$. The final state will depend on how you added the heat $\Delta Q$. Adding the heat $\Delta Q$ in a reversible process will result in different values for $P_2, V_2, T_2$ compared with ...


2

Noether's theorem states that if a system has a continuous symmetry, there is a quantity related to this symmetry, called the Noether charge, which is conserved. It does not state anything on the fact that adding a constant term to a measurable quantity may or may not change the physical description of the system. Only some physical quantities in fact are ...


1

Really how does the entropy of the universe increase? The statement "entropy of the universe increases" is a misconception about thermodynamics. The problem with that statement, as CuriousOne has written in the comments, is that universe is not a closed system amenable to thermodynamic description. It has no volume, no temperature and we have no means ...


1

I am not sure if this answer will give you an intuitive understanding of the result, but I think it may be useful as it shows the assumptions behind it. What your result means is that in an idealized situation when the volume gas or solute occupies is shrunk slightly by $\delta << V$ while its energy remains the same (let's say, isothermal compression ...


1

Boltzmann's formula for entropy is S = k * log W, where S is entropy, k is a constant, and W is the total number of ways the micro particles of a system can be re-arranged without altering the macro appearance and properties of the system. What Boltzmann had in mind was a gas tending toward thermodynamic equilibrium. For any other system, the measurement ...


1

This would depend on what entropy of plant and animal is supposed to mean. Originally, entropy describes systems in states of thermodynamic equilibrium. If you want to introduce similar quantity for systems in more complicated states, (plant or animal are not systems in thermodynamic equilibrium), you need to give its definition. There is no universally ...


1

Lie down and fall asleep. Right on the razor's edge between awake and asleep, look at what it all looks like. See yourself and the world in unison, fading away... slowly, and at the same time instantaniously. Then, in the morning, be thankful that your experience was merely that of falling asleep, and not a more permanent heat-death. (The question is ...


1

For an infinitesimal heat transfer $\delta Q$ the inequality of Clausius states that $\Delta S = S_1-S_0 = \int_0^1 {\dfrac {\delta Q_\text{rev}}{T}} > \int_0^1 {\dfrac {\delta Q_\text{irrev}}{T}}$ Here $\delta Q_\text{rev}$ and $\delta Q_\text{irrev}$ denote reversible and irreversible heat transfers, respectively. Thus if the process is reversible ...


1

A somewhat longer answer, since I'm afraid my comment may have seemed a bit abrupt... Lets look at a fairly simple thermodynamic system, the Ag-Ge binary phase diagram. This consists of 3 phases only, fcc Ag, diamond cubic Ge, and the liquid. Taking the published thermodynamic model from J. Wang et al. in Thermochimica Acta 512 240-246 (2011), one can ...



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