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You're typically talking about a "container of gas" kind of system, right? So consider a toy model where that container contains just two molecules of masses $m_1$ and $m_2$ with velocities $\vec v_1$ and $\vec v_2$, respectively. So the total internal energy of our system is $U=\frac12m_1v_1^2+\frac12m_2v_2^2$ (ignoring any internal degrees of freedom of ...

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Your question is how the differential quotient $\frac {dU}{dS}$ can mean anything in equilibrium when the quantities $U$ and $S$ are supposed to be constant, it is equilibrium after all... Indeed, $dU$ or $dS$ do not mean changes over time in a physical sense, ie., over time during some process, Instead they mean the differentials of the respective ...

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It constantly gets away from equilibrium, it just has a very low probability of getting away far enough to be measurable. In my mind, equilibrium is maximal entropy, so I'm having difficulty understanding the question. To correct an error: the energy U does not change in an isolated system.

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Lets not complicate it with the use of thermodynamic terms.Temperature is a measure of the internal energy of a substance and internal energy is related with the change is the heat which is turns relate with entropy change.

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How do we define temperature? You have to start with the zeroth law of thermodynamics which is all about bodies in thermal equilibrium. The strange name is because after the first and second laws of thermodynamics were formulated suddenly somebody realised there was another law of thermodynamics which in some ways was more fundamental than one and two, so ...

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You say: We shouldn't care about how we reached that equilibrium In fact the entire point of the example is that we do. I shall try to explain why. Jaynes and Gull both work in the framework of Bayesian inference (I can recommend the introductory text: http://www.amazon.co.uk/Data-Analysis-A-Bayesian-Tutorial/dp/0198568320. The title may seem ...

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While a funny-looking coincidence, this is not a valid alternative expression for entropy in general, since the entropy of a probability distribution (which are what rigorously hides behind the strange word "macrostate") is more generally given by $$S = - k_B \sum_i p_i\ln(p_i) \tag{1}$$ and becomes only $$S = k_B \ln(\Omega) \tag{2}$$ in the case of a ...

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Let us start with an example, called Langevin paramagnetism, where the magnetic moment is described classically, as a vector in three dimensions. Calling $\vec\mu$ this moment, $\vec B$ the magnetic induction and $\theta$ the angle between $\vec\mu$ and $\vec B$. The probability density of the angle $\theta$ is $\rho(\theta)=\frac{1}{\mathcal Z}\exp(\beta\mu ... 0 The main reason sum of countable set instead of integral over continuous volume is introduced in statistical physics in the context of classical physics is simplification of mathematics needed to make use of the probability theory to explain basics of information theory and derive the very notion of information entropy, given by the$p\ln p$formula. Or use ... 1 From a mathematical perspective this means that it is not differentiable. The problem is that you need the discreteness to be able to count states. If you replace the discreteness by something smooth you get something differentiable, but your definition of entropy no longer makes sense. This is just one of the points where mathematicians cringe, but it works ... 1 It depends what you mean by immortal: If it means to transcend time, can anything wholly mortal or finite transcend time? Even in religous philosophies already some divine element, spark or breath ie something transcendent or immortal in this sense is mixed in with the human clay ie mortal or finite. If it means to to last as long as time lasts - at least ... 4 Please note that the second law of thermodynamics applies to isolated systems. Life, the simplest cell, is an engine for reducing entropy in its volume and increasing entropy in its surroundings, so life is not limited by entropy because it develops within a larger medium, which can be considered as isolated. Crystals also do the same, when something ... 0 Your question does not make a lot of sense, which is due to the words "microstate" and "macrostate" floating around without being given a precise meaning. A microstate is a point in the phase space of the system - it refers to a single, unambiguous configuration of the system. For a system of$N$freely moving particles (an ideal gas) the phase space is ... 0 Citing Wikipedia here, In what has been called the fundamental assumption of statistical thermodynamics or the fundamental postulate in statistical mechanics, the occupation of any microstate is assumed to be equally probable (i.e. Pi = 1/Ω, where Ω is the number of microstates); this assumption is usually justified for an isolated system in ... 1 A macrostate is a set of microstates. Some microstates are thermal, others are not. Without the assumption of being in thermal equilibrium you can't assume anything since any possible microstate is possible. And lots of possibilities macrostates could be picked. Usually you want to group your macrostates according to a state variables such as pressure, ... -1 Of course the order in an open system always decreases. Locally, like on the Earth, beautifull orderly structures can evolve (wich wouldn´t be possible if the Earth always had one side directed to the Sun), and nonetheless the total order of the system Sun-Earth would decrease. But human activity, by wich I mean the changing of the surface of the Earth by ... 0 In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations. In contrast, the macrostate of a system refers to its macroscopic properties, such as its temperature, pressure, volume and density. Treatments on ... 1 The second law of thermodynamics says that order always decreases in a system. This is incorrect. My air conditioner alone proves that this statement of the second law of thermodynamics is incorrect. I live in Houston, TX; I rely on my air conditioner for about six months of the year. My air conditioner does not violate the second law of thermodynamics. ... 2 As mentioned in a comment, entropy cannot decrease in a closed system. When you are considered life on Earth, it is not a closed system. Energy comes into the earth from the sun and radiates away from the earth as heat. In thermodynamics, we normally consider mostly engines because they are really simple in comparison to other systems. If we ignore the ... 2 The equilibrium thermodynamic state of a single phase substance of constant composition is determined by specifying two intensive properties. Note that, from the equation of state, V and T uniquely determine P (where V is the specific volume). 5 For any function of multiple variables, say,$f(x, y, z, \ldots)$you can always write the following:$df = \big( \frac{\partial f}{\partial x} \big)_{y, z, \ldots} dx + \big( \frac{\partial f}{\partial y} \big)_{x, z, \ldots} dy + \big( \frac{\partial f}{\partial z} \big)_{x, y, \ldots} dz + \ldots$In fact, the constancy of the other variables is ... 0 No actually this is one perpetuating myth about entropy that even scientists themselves (and school curricula) propagate. To answer this and dispel the myth, ask this simple question: disorder with respect to what exactly? Why is a uniform gas disordered than a gas with two phases? Of course a uniform gas has more (another) symmetry, in fact aquires the ... 1 The entropy law can be (comically) reinterpreted like "equilibrium is a state of maximum possible disorder under given physical constraints". So... things keep getting worse until it's as bad as it can get. Intuitively, large entropy means that things look more or less the same (macroscopically) for many different microscopic realizations. When the system ... 3 Entropy is not disorder; it is a lack of information. Consider the entropy formula$S = k_b \log \Omega$. Here,$\Omega$is the number of microstates (sets of particle positions/momenta) corresponding to an observed macrostate (something macroscopic we can observe, like 'the gas has volume$V$and pressure$P$). What this formula means is that the entropy ... 0 Entropy is a tricky concept and hard to understand. Personally I tend to avoid speaking of systems and phenomena in terms of entropy and/or temperature because they say very little of the dynamics, and I believe dynamical laws are the ones driving the universe. When we hear that systems tend to increase entropy, we are saying there are dynamical laws ... 8 What you are missing is the microscopic definition of entropy, once you know that, you will understand why people say that entropy is disorder. Equilibrium as order First, let's address your valid intuition that equilibrium as a form of order. Indeed, if everything is in thermal equilibrium, you just need to measure the temperature somewhere, and then you ... 6 First of all as stated by Madan Ivan: equilibrium is not order. But you can get certain systems that are in a meta-stable "local" equilibrium (here meaning that you need some energy to move it from there), for example a crystal. These can be highly ordered. Intuitively: if you smack the crystal with a hammer it breaks to pieces. This brings your closer to ... 13 I personally find the terms consistent. Think of the entropy as Boltzman proposes:$S=k \, \ln W$Meaning high entropy states can be realized via many different configurations. Truly ordered state (assume you arrange a sculpture from atoms) can be realized via much smaller number of microscopic states. So again, equilibrium is not order - it is a mess. -2 Terms are conventions. With a point of view from the humans we are the order. Collecting something and order it in shells is order. But I agree with you that to order something needs energy and this led to misorder and this could be a possible convention too. But it is not. 0 Suppose that a system is subjected to an irreversible process between two thermodynamic equilibrium states. To determine the change in entropy of the system$\Delta S\$ between these two equilibrium states, you execute the following sequence of steps: Forget about the irreversible process path. It cannot be used to determine the change in entropy. Focus ...

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Time is just a scale we use to measure the rate of processes or to measure the interval between 2 events. Time is not a stand-alone entity and it does not exist alone independently. All the means (like clocks) we employ to measure Time use some standard physical changes as their fundamental measuring units of Time. So Time has no direction of its own, ...

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What I try to say, is that it makes equally sense if a ball rolls down of tray or up (without friction) in a newtonian system. The "proces" is then reversible. In thermodynamics a proces will only go one way in a given situation therefore irreversible. – Hamid Mohammad 18 hours ago When you think of a ball rolling up or down a hill it is easy to ...

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You have to be a bit careful about what you mean when you ask about Newtonian mechanics being reversible or not. As stated in one of the comments newtons mechanics is only a set of rules that tell you how objects accelerate if they are subject to some sets of forces. It does not necessarily tell you the nature of these forces of where they come from. To ...

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This is a question that cannot be answered without understanding what time is. And I think the best we can say is that time is a quantity we use to compare events. But the compared feature is the ordering. Bearing that in mind, we cannot say much about its reality, even less about its direction. In other words, despite the fact that you can order events, ...

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Time is relative, meaning that time never runs exactly the same in 1 place than another even on the same world. Time runs fast next to the sun while time on Pluto would run slower than here. Next to a black hole time would from almost seam to stop. There is no universal clock to set time by only what we created on Earth for ourselves. Time only moves ...

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The reason why heat cannot flow from cold to warm is that the change in entropy will become negative, and that doesn't happen in a closed system. Negative entropy is by definition not possible. Here's why: Alternative example: gas in a box I think entropy gets a little more intuitive if we think of it in terms of statistical mechanics. If we imagine a box ...

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The 2nd Law of Thermodynamics is based on an overwhelmingly extensive body of empirical evidence on how thermodynamic systems behave. There are many different statements of the 2nd Law, and all of them are equivalent to one another. Once one of these has been specified, all the other follow. One such statement says that heat cannot flow spontaneously from ...

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