Hot answers tagged

8

I would say the connection between chaos and entropy is through ergodic theory, and the fundamental assumption of statistical mechanics that a system with a given energy is equally likely to be found in any 'microstate' with that energy. Although chaos is a very general aspect of dynamical systems, Hamiltonian chaos (encountered in classical mechanics) is ...


7

Formally, the two entropies are the same thing. The Gibbs entropy, in thermodynamics, is $$S = -k_B \sum p_i \ln p_i$$ while the Shannon entropy of information theory is $$H = -\sum p_i \log_2 p_i.$$ These are equal up to some numerical factors. Given a statistical ensemble, you can calculate its (thermodynamic) entropy using the Shannon entropy, then ...


5

The Boltzmann brain paradox arises due to smaller fluctuations being more probable than larger ones. So, if you contemplate how our universe started out with low entropy initial conditions then it's difficult to explain this in terms of a generic high entropy state. Fluctuation yielding the early universe that in turn would have given rise to you, would be ...


4

If the phase change occurs at the temperature of interest, then the system can give off a lot of heat without cooling down very much. Thus, melting ice is a great way to maintain something at a temperature around 0°C, and melting paraffin-18-Carbons is good if you are trying to maintain temperature around 20 °C. With a melting point of 28°C, the material ...


4

Gravity can appear to increase order, but it doesn't violate the second law of thermodynamics. For example, take a gas cloud at uniform density and temperature $T_1$ as our system. If we let it condense under its own gravity we can consider two cases: (1) Assuming it forms a gas giant quickly without radiating any heat (i.e. adiabatically), it will have a ...


4

I'll have to disagree that those notions of entropy are disjoint. I'll try to explain my view. In Statistical Mechanics entropy is defined in terms of accessible regions in phase space. It is the logarithm of this volume times a constant. In the process of deriving this formula starting from the number of accessible configurations it is postulated that all ...


3

Entropy Demystified (The Second Law Reduced to Plain Common Sense) by Arieh Ben-Naim. Authored discussed not only the thermodynamics origin of entropy but also the same notion in the context of information theory developed by Claude Shannon.


3

Is this taken to be an additional (and apparently implicit) assumption? You are correct. Take two arbitrary points $A,B$ on the $PV$ (or any other) plane, and draw an arbitrary curve connecting them: you have just defined a reversible transformation connecting $A$ and $B$. This is because every point in the $PV$ (or any other) plane represents an ...


3

So Pratchett's quote seems to be about energy, rather than entropy. I supposed you could claim otherwise if you assume "entropy is knowledge," but I think that's exactly backwards: I think that knowledge is a special case of low entropy. But your question is still interesting. The entropy $S$ in thermodynamics is related to the number of indistinguishable ...


3

We know that: irreversible+adiabatic = $\Delta S>0$, thus, if $\Delta S=0$ the process is either: irreversible+non-adiabatic, reversible+adiabatic, or reversible+non-adiabatic. Then you can conclude that: $\Delta S=0$+adiabatic=reversible. Regarding you example of an irreversible adiabatic cycle: It is impossible and the example is flawed. The ...


3

So far there have been quite a few insightful answers about statistical mechanical entropy, but so far the only mention of thermodynamic entropy has been made by CuriousOne in the comments, so I thought it would be useful to give a short general reminder about the subtle difference between the notion of entropy in thermodynamics and the formulas that come up ...


3

The hamiltonian of a perfect crystal can be approximated at low temperature as the sum of harmonic oscillator hamiltonians. In 1D we have $$H = \sum_{i=1}^N \frac{p_i^2}{2 m} + \frac 1 2 m \omega^2 \sum_{ij} ( r_i- r_j)^2 $$ where the $ij$ sum is over nearest neighbors. It is possible to verify that the eigenvalues of this hamiltonian are $$ E_n = \left( ...


3

I would like to answer with the words of L.D. Landau, from his book Statistical Physics (first edition $1958$):


2

An intuitive way of thinking about boiling an egg, is that you start a pool with a bunch of balls of yarn floating in it. Then when you heat the egg, it causes all the balls of yarn to unravel, and rather than a pool with some balls in it now it's a tangled mess of yarn. Before, each ball was free to move around as a unit, now each segment of each strand can ...


2

What is meant by a “change in volume of a system”? "Change in volume of a system" means "change in volume of a system", not anything else. System is a hypothetical concept. There is no specified system before we define it. We ourselves choose and define system. When someone talks about a system defined by himself/herself, he/she talks about that system not ...


2

From the links I provided in the comments below your question it should become clear that entropy "meters" do not exist, you calculate it from other measured variables. If this does not satisfy you requisites for an experimental measurement, then your conclusion that the claims are only theoretical is justified. However, having said that, with your ...


2

Leaving aside the issue of wavefunction collapse, physics is deterministic. So if you have some system like a gas and you know the exact positions and velocities of all the gas molecules you can predict the evolution of the system forwards and backwards in time. So you can start with a future state and work backwards to desciribe a past state. However ...


2

To be honest, I believe this question is not really settled, or at least that there is not yet a consensus in the scientific community about what the answer is. My understanding of the relation is, I think, slightly different than knzhou, rob, or CuriousOne. My understanding is that thermodynamic entropy can be thought of as a particular application of ...


2

A Boltzmann brain is not that different from what might be called Boltzmann cheese. Given enough time a set of atoms or particles might arrange themselves by statistical fluctuations into big wheel of cheese. If that happens there is no reason to think the cheese would then rapidly be demolished unless it formed in a star, or falling into a black hole or in ...


1

So, you want to prove that between any arbitrary two states of a system, it exists at least one reversible path. You can prove this if you accept continuity of properties of substances. I.e. for example, if we have an ideal gas in equilibrium at initial state $(P_i,T_i)$ and final state $(P_f,T_f)$; then certainly there are infinite equilibrium states ...


1

If you can take the environment to be one with constant specific heats $c_p$ and $c_V$, then you can write $$dS_{\textrm{env}}=\frac{\delta Q_{\textrm{env}}}{T_{\textrm{env}}} = \begin{cases} \frac{nc_VdT}{T} & \textrm{if isochoric} \\ \frac{nc_pdT}{T} & \textrm{if isobaric} \end{cases}, $$ where the temperatures all refer to the environment. Then, ...


1

Your analysis is totally correct (except for the sign of the change), provided cv refer and cp refer to the heat capacities of the environment, and Tf and Ti are the final and initial temperatures of the environment. In such an analysis, the "environment" becomes your system. The signs are, of course, incorrect. They should be +'s.


1

Is this true for every reversible cycle? Is the efficiency of all reversible cycle equal to the efficiency of a Carnot Cycle? Yes. They are indeed. The equality in Clausius' Inequality $$\oint \frac{đq_\textrm{sys}}{T_\textrm{source}}=0$$ is strictly valid for all reversible cycles. Temperature of a reversible engine is at all times equal to the ...


1

In order to input in your terms 500rpm you are required to input the actual shaft rpm Plus 500 you seek to add. You must charge your input to the systems energy level. So to answer your question yes your input of 500rpm would slow down the shaft rpm.


1

Tl,dr: Entropy is the right definition, because it's incredibly useful in the description of statistical and thermodynamic systems. Whether or not it quantifies "disorder" in whatever sense of the word is completely irrelevant - it just so happens that it can be interpreted that way. Entropy is not a measure of disorder. At least not really. Then again, ...


1

The problem that I see with the idea is that it seems contradictory. You used the definition of entropy of the microcanonical ensemble, which is defined by assigning an equal probability to every microstate whose energy falls within a range centered at E. All other microstates are given a probability of zero. That is, the range of energy is reduced in width ...


1

More of an extended comment, but here are two thoughts: 1) I'm not sure I completely agree with the statement 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. Okay, this is correct, of course, ...


1

On top of the other excellent answers I'd like to point out that the accretion rate of dark matter particles is believed to be much smaller. The reason matter in accretion disk is being accreted rapidly is because they lose energy from electromagnetic radiation. For dark matter particles, in practice the only way it can be accreted is if the particle ...


1

Here are some references: Time and chance by david Albert Time's arrow by huw price From eternity to here by sean carroll The direction of time by H. D. Zeh Physical basis of time Asymmetry by paul davies There are many other excellent books or articles about the subject. Especially, in relation to the foundation of statistical mechanics I saw Two ...


1

So that narrows it down to some time between 1896 and 1979 The second law was known to Clausius, and trivially implies the knowledge that the entropy in the far past was much less than now. (That is, if one is permitted to apply the notion to the universe as a whole; cf. below.) It seems that Clausius stated explicitly (in 1856) only the extrapolation to ...



Only top voted, non community-wiki answers of a minimum length are eligible