I'm reading through Kardar's Statistical Mechanics of Particles; in the section 1.5 he says:     A reversible process is one that can be run backward in time by simply reversing its inputs and outputs. It is the thermodynamic equivalent of frictionless motion in mechanics. Since time reversibility implies equilibrium, a reversible transformation must be quasi-static, but the reverse is not necessarily true.   Why time reversibility implies equilibrium ?
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A process is reversible if and only if it's always at equilibrium during the process.  Why?  I have heard several specific example of this, such as adding weight gradually to a piston to compress the air inside reversibly, by why should it be true in general?  EDIT: Here is something that would firmly convince me of this: Suppose I have a reversible process that is not always in equilibrium.  Describe a mechanism for exploiting this process to create a perpetual motion machine.
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In classical mechanics, the Hamiltonian is well defined by the Lagrangian. Whereas, energy is a very ambiguous term. We just say $E=T+U$, and usually it equals to Hamiltonian. Does there exist a way that, by just looking at the Lagrangian mathematically, we immediately know the relationship between the Hamiltonian and the energy of the system?  And if we have a system, the Hamiltonian of which does not equal to energy, what is the physical meaning of that difference?
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Examples from Principles of Physics (by Walker,Resnick,Halliday) will say it better:     Let us throw a tomato upward. . .as the tomato rises, the work $\mathbf{W_g}$ done on the tomato by the gravitational force is negative as the force transfers energy from the $E_K$ of the tomato. We can now finish the story by saying that this energy is transferred by gravitational force to the gravitational potential energy of the tomato-earth system.   The tomato slows down, stops &amp; then begins to fall back down. During the fall, the transfer is reversed: the work $\mathbf{W_g}$ done on tomato by the gravitational force is now positive - that force transfers energy from the gravitational potential energy of the tomato-Earth system to the $E_K$ of the tomato.      If we abruptly shove a block to send it moving rightward, the spring force acts leftward &amp; thus does negative work on the block, transferring energy from the $E_K$ of the block to the elastic potential energy of the spring-block system. . . .   And many,many other examples. Now, why is the potential energy attributed to the system and the kinetic energy to the object itself? What is the logic/cause?
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I'm looking for an example of a Hamiltonian $H$, where $H\neq T+V$, but the total energy in the system, $E=T+V$, is still conserved.   While I'm at it, I might as well add that I'd be most interested in an example from a classical field theory. Furthermore, I am looking for a nontrivial example, e.g. where $H$ doesn't just differ from $E$ by some gradient function that integrates to 0 in the action.   In particular, I am trying to get a better understanding of the physical implications of $H$ being conserved when nontrivial variables are used, i.e. physically, what does $H$ this correspond to. Conversely, can one have a system where $E = H$ in one frame, and is conserved, while in another frame $E'\neq H'$, and $H'$ is conserved while $E'$ is not?
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I guess I understand the equation of the detailed balance condition, which states that for transition probability $q$ and stationary distribution $\pi$, a Markov Chain satisfies detailed balance if $$q(x|y)\pi(y)=q(y|x)\pi(x),$$  this makes more sense to me if I restate it as:   $$\frac{q(x|y)}{q(y|x)}= \frac{\pi(x)}{\pi(y)}. $$  Basically, the probability of transition from state $x$ to state $y$ should be proportional to the ratio of their probability densities.
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Let $X$ be a discrete-time Markov process on some measurable space $(\mathscr X,\mathscr B)$. Let $B\in\mathscr B$ and  $$ \tau_B:=\inf\{n\geq 0:X_n\in B\} $$ is the first hitting time of $B$.    Let us assume that for any $x\in \mathscr X$  $$ \mathsf P_x\{\tau_B&lt;\infty\} = 1. $$ Can there exist $x$ such that $\mathsf E_{x}[\tau_B] = \infty$? Let us assume that for some $x&#39;\in \mathscr X$  $$ \mathsf P_{x&#39;}\{\tau_B&lt;\infty\} = 1. $$ Can it be that $\mathsf E_{x&#39;}[\tau_B] = \infty$?   No assumptions on the irreducibility (or any other properties) are imposed. I've tried to find a counterexample but didn't manage to do it.
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In Sean Carroll's book "The Big Picture," he states ():      Classical mechanics, the system of equations studied by Newton and   Laplace, isn't perfectly deterministic. There are examples of cases where a unique outcome cannot be predicted from the current state of the system. This doesn't bother most people, since cases like this are extremely rare—they are essentially infinitely unlikely among the set of all possible things a system could be doing. They are artificial and fun to think about, but not of great import to what happens in the messy world around us.   What are some of these situations he is referring to?   My first guess was something like the conceptual equivalent of a ball sitting exactly atop a hill and asking which direction it will roll. But that doesn't seem right—this would be a case where the "true" solution is that the ball wouldn't roll at all, while in reality there will always be tiny disturbances that cause the ball to roll in some direction. I don't think this is what Carroll is referring to. He seems to suggest that the "true, exact mathematical outcome" is not uniquely determined for certain classical systems.  I've heard that uniqueness theorems for solutions to differential equations don't generally apply to non-linear equations, and wonder if this may have something to do with it. Elucidation and examples would be much appreciated.
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Could someone provide me with a mathematical proof of why, a system with an absolute  (such that of a spin system) is  than any system with a positive temperature (in the sense that if a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system).
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Schroedinger's equation, as we all know, is time symmetric. In quantum field theory, we have to come up with a more sophisticated CPT reversal, but the essential point remains unchanged. However, the collapse of the wave function in the Copenhagen interpretation is manifestly time asymmetric. Correct me if I'm wrong, but can you uncollapse a wave function, converting it from an eigenfunction to a superposition of eigenfunctions?  Is this asymmetry connected with the thermodynamic arrow of time and the second law in statistical mechanics, or are they independent? How would an uncollapse look like, and can we experimentally arrange for an uncollapse? Why are there more collapses than uncollapses? If an observer unobserves a quantum thingie, does that thingey uncollapse?
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According to 2nd law of thermodynamics, it is not possible to convert heat 100% into work.   Well, investigating about the Carnot Cycle, I have found out that in the first step, the process is to heat a gas without rising its temperature, this is, converting all the heat into work. Isn't this a violation of the 2nd law?  I clearly don't understand something about the law or about the Carnot Cycle. I would like to know where I'm failing in my understandig. Thank you.
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The second law says that entropy can only increase, and entropy is proportional to phase space volume. But Liouville's theorem says that phase space volume is constant.  Taken naively, this seems to imply that the entropy can never change. What's wrong with the reasoning here?
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new finding by Minev et al. seems to suggest that transitions between atomic states are not instantaneous, but continuous processes wherein a superposition smoothly adjusts from favoring one state to another (if I understand it correctly). The authors also claim to be able to catch a system "mid-jump" and reverse it. Popular articles are  and .  I am curious if this finding rules out any interpretations of QM. It seems to generally go against the Copenhagen attitude, which describes measurements as collapsing physical systems into a definite classical state. The popular articles indeed claim that the founders of QM would have been surprised by the new finding.  The link with the asterisk mentions that something called "quantum trajectories theory" predicts what was observed. Is this an interpretation, or a theory? And are they implying that other interpretations/theories don't work?
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The question is sometimes referred to as the "psychological arrow of time" (Hawking, 1985). Here the past is understood as a moment or time when the entropy of the universe was lower, and contrarily for the future. So it is generally thought that PAOT is a consequence of the thermodynamic arrow of time of our universe. If so (maybe not?), how do the two relate?   Some explanations in the literature:   Practical memory systems work in a way that the formation of new memories entails an overall increase of total entropy of the system and the environment. For example, to create a memory, i.e. to cause our neurons to orient in a particular fashion, requires energy which results in our body heating up a little bit, increasing the total entropy (Hawking, 1985 and 1994); The initialization of memory to make it reusable is an irreversible process that increases total entropy (Landauer, 1961. Wolpert, 1992). More recently, People have argued that even reversible and non-dissipative memory systems are subject to PAOT (Mlodinow and Brun, 2014). The conclusion is arrived by imposing some constraints on what a memory system should be like. Specifically, they argue that a memory should be somehow robust to small microscopic changes in states of the system it records (what they call "generality" requirement). But the smallest changes in the future state destroy the thermodynamic arrow of time between now and the future. So any memory of the future of the system "could remember only one possible configuration of that system". This fine-tuning disqualifies it as a bona fide memory.   My problem with explanation (1) is that even if it's correct, it doesn't seem to be a complete answer in itself. Yes, increase of (new) memory happens only as total entropy of the universe increases. So what? It doesn't have anything to say on the nature of that memory. Why couldn't it occasionally be a memory of the future for that matter? Explanation (2) leaves no such ambiguity. But the generality requirement seems artificial: surely a memory that records the only future configuration of the system remembers the future in a deterministic world, there being no "what ifs" regarding that state?   Of course, my understanding of the problem is only preliminary. I'd like to know whether there is not some generally accepted explanation, or any other thoughts you have on it.
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I know that taking the derivative of position with respect to time defines what we call velocity, but I've never heard of physicist going in the opposite direction with position. Is there any application for the integral of position with respect to time?  Maybe with respect to something else?
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Consider a heat reservoir which gains heat $Q$ irreversibly at temperature $T$ from the surroundings which is at temperature $T_0$. The entropy change of reservoir is then given by $\frac{Q}{T}$, while that of the surroundings is $-\frac{Q}{T_0}$.  My question is, how is this possible? According to the Clausius inequality, the entropy change of a irreversible process is greater than that due to heat transfer. Please help, thank you!
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I have some confusion about the "Reversed effective force" as it appears in the derivation of .  In Goldstein d'Alembert's principle is given as:  $(F-\dot{p}) \cdot \delta r = 0$  First I have sources that seem to be contradictory.    This book, on page 8, indicates that "reversed effective forces" are not real forces.  However, this book indicates otherwise.    On page 345, this book indicates "this is the force exerted by the moving body to resist the change in its state.  I think that the former source is probably correct, but I lack good intuition of D'Alembert's principle. For reference the reversed effective force is represented by $-ma$ or $-\dot{p}$. My best guess as to what the reversed effective force is that it it the force that is required to appear if you view the system from the perspective of the accelerating body.  Is this correct?  If it is then how is d'Alembert's principle applicable to inertial systems?
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How do you prove the  from statistical mechanics? To prove entropy will only increase with time? How to prove? Please guide.