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# definitions - Second_law_of_thermodynamics

## second law of thermodynamics(n.)

1.a law stating that mechanical work can be derived from a body only when that body interacts with another at a lower temperature; any spontaneous process results in an increase of entropy

# definition (more)

definition of Wikipedia

# analogical dictionary

second law of thermodynamics (n.)

# Second law of thermodynamics

In classical thermodynamics, the laws of thermodynamics are basic postulates applicable to any system involving measurable heat transfer. In statistical thermodynamics, the second law is a consequence of unitarity in quantum theory. In classical thermodynamics, however, the second law defines the concept of thermodynamic entropy, while in statistical mechanics entropy is defined from information theory, known as the Shannon entropy. In such instances, the second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system so as to result in the natural entropic dissolution of the system itself. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and explains the phenomenon of irreversibility in nature. The second law declares the impossibility of machines that generate usable energy from the abundant internal energy of nature by processes called perpetual motion of the second kind.

The second law may be expressed in many specific ways, but the first formulation is credited to the French scientist Sadi Carnot in 1824 (see Timeline of thermodynamics). The law is usually stated in physical terms of impossible processes.

## Description

The first law of thermodynamics provides the basic definition of thermodynamic energy, also called internal energy, associated with all thermodynamic systems, but unknown in mechanics, and states the rule of conservation of energy in nature.

However, the concept of energy in the first law does not account for the observation that natural processes have a preferred direction of progress. For example, spontaneously, heat always flows to regions of lower temperature, never to regions of higher temperature without external work being performed on the system. The first law is completely symmetrical with respect to the initial and final states of an evolving system. The key concept for the explanation of this phenomenon through the second law of thermodynamics is the definition of a new physical property, the entropy.

A change in the entropy (S) of a system is the infinitesimal transfer of heat (δQ) to a closed system driving a reversible process, divided by the equilibrium temperature (T) of the system.[1]

$dS = \frac{\delta Q}{T} \!$

The entropy of an isolated system that is in equilibrium is constant and has reached its maximum value.

Empirical temperature and its scale is usually defined on the principles of thermodynamics equilibrium by the zeroth law of thermodynamics.[2] However, based on the entropy, the second law permits a definition of the absolute, thermodynamic temperature, which has its null point at absolute zero.[3]

The second law of thermodynamics may be expressed in many specific ways,[4] the most prominent classical statements[3] being the statement by Rudolph Clausius (1850), the formulation by Lord Kelvin (1851), and the definition in axiomatic thermodynamics by Constantin Carathéodory (1909). These statements cast the law in general physical terms citing the impossibility of certain processes. They have been shown to be equivalent.

### Clausius statement

German scientist Rudolf Clausius is credited with the first formulation of the second law, now known as the Clausius statement:[4]

No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.[note 1]

Spontaneously, heat cannot flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, a compressor.

### Kelvin statement

Lord Kelvin expressed the second law in another form. The Kelvin statement expresses it as follows:[4]

No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work.

This means it is impossible to extract energy by heat from a high-temperature energy source and then convert all of the energy into work. At least some of the energy must be passed on to heat a low-temperature energy sink. Thus, a heat engine with 100% efficiency is thermodynamically impossible. This also means that it is impossible to build solar panels that generate electricity solely from the infrared band of the electromagnetic spectrum without consideration of the temperature on the other side of the panel (as is the case with conventional solar panels that operate in the visible spectrum).

Note that it is possible to convert heat completely into work, such as the isothermal expansion of ideal gas. However, such a process has an additional result. In the case of the isothermal expansion, the volume of the gas increases and never goes back without outside interference.

### Principle of Carathéodory

Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:[5]

In every neighborhood of any state S of an adiabatically isolated system there are states inaccessible from S.[6]

With this formulation he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics.

### Equivalence of the statements

Derive Kelvin Statement from Clausius Statement

Suppose there is an engine violating the Kelvin statement: i.e.,one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the graph. The net and sole effect of this newly created engine consisting of the two engines mentioned is transferring heat $\Delta Q=Q\left(\frac{1}{\eta}-1\right)$ from the cooler reservoir to the hotter one, which violates the Clausius statement. Thus a violation of the Kelvin statement implies a violation of the Clausius statement, i.e. the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, and hence the two are equivalent.

## Corollaries

### Perpetual motion of the second kind

Prior to the establishment of the Second Law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of First Law of Thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.

### Carnot theorem

Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:

• All irreversible heat engines between two heat reservoirs are less efficient than a Carnot engine operating between the same reservoirs.
• All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.

In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot however further postulated that some caloric is lost, not being converted to mechanical work. Hence no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.

Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.

### Clausius Inequality

The Clausius Theorem (1854) states that in a cyclic process

$\oint \frac{\delta Q}{T} \leq 0.$

The equality holds in the reversible case[7] and the '<' is in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero.

### Thermodynamic temperature

For an arbitrary heat engine, the efficiency is:

$\eta = \frac {A}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)$

where A is the work done per cycle. Thus the efficiency depends only on qC/qH.

Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures T1 and T2 must have the same efficiency, that is to say, the effiency is the function of temperatures only: $\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).$

In addition, a reversible heat engine operating between temperatures T1 and T3 must have the same efficiency as one consisting of two cycles, one between T1 and another (intermediate) temperature T2, and the second between T2 andT3. This can only be the case if

$f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).$

Now consider the case where $T_1$ is a fixed reference temperature: the temperature of the triple point of water. Then for any T2 and T3,

$f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}.$

Therefore if thermodynamic temperature is defined by

$T = 273.16 \cdot f(T_1,T) \,$

then the function f, viewed as a function of thermodynamic temperature, is simply

$f(T_2,T_3) = \frac{T_3}{T_2},$

and the reference temperature T1 will have the value 273.16. (Of course any reference temperature and any positive numerical value could be used—the choice here corresponds to the Kelvin scale.)

### Entropy

According to the Clausius equality, for a reversible process

$\oint \frac{\delta Q}{T}=0$

That means the line integral $\int_L \frac{\delta Q}{T}$ is path independent.

So we can define a state function S called entropy, which satisfies

$dS = \frac{\delta Q}{T} \!$

With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the Third Law of Thermodynamics, which states that S=0 at absolute zero for perfect crystals.

For any irreversible process, since entropy is a state function, we can always connect the initial and terminal status with an imaginary reversible process and integrating on that path to calculate the difference in entropy.

Now reverse the reversible process and combine it with the said irreversible process. Applying Clausius inequality on this loop,

$-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0$

Thus,

$\Delta S \ge \int \frac{\delta Q}{T} \,\!$

where the equality holds if the transformation is reversible.

Notice that if the process is an adiabatic process, then $\delta Q=0$, so $\Delta S\ge 0$.

### Available useful work

An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature TR and pressure PR — so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain TR; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain PR.

Whatever changes to dS and dSR occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy Stot of the isolated total system must not decrease:

$DS_{\mathrm{tot}}= dS + dS_R \ge 0$

According to the First Law of Thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μiRNi, so that:

$DU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \,$

where μiR are the chemical potentials of chemical species in the external surroundings.

Now the heat leaving the reservoir and entering the sub-system is

$\delta q = T_R (-dS_R) \le T_R dS$

where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.

It therefore follows that any net work δw done by the sub-system must obey

$\delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \,$

It is useful to separate the work δw done by the subsystem into the useful work δwu that can be done by the sub-system, over and beyond the work pR dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work that can be done:

$\delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\,$

It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy X of the subsystem,

$X = U - T_R S + p_R V - \sum \mu_{iR} N_i$

The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,

$D X + \delta w_u \le 0 \,$

i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.

In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in X for an irreversible process and no change for a reversible process.

$dS_{tot} \ge 0$ Is equivalent to $dX + \delta W_u \le 0$

This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)

This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.

## History

Nicolas Léonard Sadi Carnot in the traditional uniform of a student of the École Polytechnique.

The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.

Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).

Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.

The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.

It has been shown that not only classical systems but also quantum mechanical ones tend to maximize their entropy over time. Thus the second law follows, given initial conditions with low entropy. More precisely, it has been shown that the local von Neumann entropy is at its maximum value with a very high probability.[8] The result is valid for a large class of isolated quantum systems (e.g. a gas in a container). While the full system is pure and therefore does not have any entropy, the entanglement between gas and container gives rise to an increase of the local entropy of the gas. This result is one of the most important achievements of quantum thermodynamics[dubious ].

Today, much effort in the field is attempting to understand why the initial conditions early in the universe were those of low entropy,[9][10] as this is seen as the origin of the second law (see below).

### Informal descriptions

The second law can be stated in various succinct ways, including:

• It is impossible to produce work in the surroundings using a cyclic process connected to a single heat reservoir (Kelvin, 1851).
• It is impossible to carry out a cyclic process using an engine connected to two heat reservoirs that will have as its only effect the transfer of a quantity of heat from the low-temperature reservoir to the high-temperature reservoir (Clausius, 1854).
• If thermodynamic work is to be done at a finite rate, free energy must be expended.[11]

### Mathematical descriptions

Rudolf Clausius

In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:[12]

$\int \frac{\delta Q}{T} = -N$

where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:

The entropy of the universe tends to a maximum.

This statement is the best-known phrasing of the second law. Moreover, owing to the general broadness of the terminology used here, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, to which this statement applies, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This, of course, is not true; this statement is only a simplified version of a more complex description.

In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:

$\frac{dS}{dt} \ge 0$

where

S is the entropy of the system and
t is time.

The equality sign holds in the case that only reversible processes take place inside the system. If irreversible processes take place (which is the case in real systems in operation) the >-sign holds. An alternative way of formulating of the second law for isolated systems is:

$\frac{dS}{dt} = \dot S_{i}$ with $\dot S_{i} \ge 0$

with $\dot S_{i}$ the sum of the rate of entropy production by all processes inside the system. The advantage of this formulation is that it shows the effect of the entropy production. The rate of entropy production is a very important concept since it determines (limits) the efficiency of thermal machines. Multiplied with ambient temperature $T_{a}$ it gives the so-called dissipated energy $P_{diss}=T_{a}\dot S_{i}$.

The expression of the second law for closed systems (so, allowing heat exchange and moving boundaries, but not exchange of matter) is:

$\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i}$ with $\dot S_{i} \ge 0$

Here

$\dot Q$ is the heat flow into the system
$T$ is the temperature at the point where the heat enters the system.

If heat is supplied to the system at several places we have to take the algebraic sum of the corresponding terms.

For open systems (also allowing exchange of matter):

$\frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i}$ with $\dot S_{i} \ge 0$

Here $\dot S$ is the flow of entropy into the system associated with the flow of matter entering the system. It should not be confused with the time derivative of the entropy. If matter is supplied at several places we have to take the algebraic sum of these contributions.

Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/√N where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.

## Derivation from statistical mechanics

In statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of $E$ is:

$S = k \log\left[\Omega\left(E\right)\right]\,$

where $\Omega\left(E\right)$ is the number of quantum states in a small interval between $E$ and $E +\delta E$. Here $\delta E$ is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of $\delta E$. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on $\delta E$.

Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then $\Omega$ will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that $\Omega$ is maximized as that is the most probable situation in equilibrium.

If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that $\Omega$ is maximized, implies that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).

The entropy of a system that is not in equilibrium can be defined as:

$S = -k_{B}\sum_{j}P_{j}\log\left(P_{j}\right)$

see here. Here the $P_{j}$ is the probabilities for the system to be found in the states labeled by the subscript j. In thermal equilibrium the probabilities for states inside the energy interval $\delta E$ are all equal to $1/\Omega$, and in that case the general definition coincides with the previous definition of S that applies to the case of thermal equilibrium.

Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number $\Omega$ of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of $1/\Omega$. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the entropy will increase continuously as a function of time during the intermediate out of equilibrium state.

### Derivation of the entropy change for reversible processes

The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:

$dS =\frac{\delta Q}{T}$

where the temperature is defined as:

$\frac{1}{k T}\equiv\beta\equiv\frac{d\log\left[\Omega\left(E\right)\right]}{dE}$

See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.

The generalized force, X, corresponding to the external variable x is defined such that $X dx$ is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate $E_{r}$ is given by:

$X = -\frac{dE_{r}}{dx}$

Since the system can be in any energy eigenstate within an interval of $\delta E$, we define the generalized force for the system as the expectation value of the above expression:

$X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,$

To evaluate the average, we partition the $\Omega\left(E\right)$ energy eigenstates by counting how many of them have a value for $\frac{dE_{r}}{dx}$ within a range between $Y$ and $Y + \delta Y$. Calling this number $\Omega_{Y}\left(E\right)$, we have:

$\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,$

The average defining the generalized force can now be written:

$X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,$

We can relate this to the derivative of the entropy w.r.t. x at constant energy E as follows. Suppose we change x to x + dx. Then $\Omega\left(E\right)$ will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between $E$ and $E+\delta E$. Let's focus again on the energy eigenstates for which $\frac{dE_{r}}{dx}$ lies within the range between $Y$ and $Y + \delta Y$. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E – Y dx to E move from below E to above E. There are

$N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,$

such energy eigenstates. If $Y dx\leq\delta E$, all these energy eigenstates will move into the range between $E$ and $E+\delta E$ and contribute to an increase in $\Omega$. The number of energy eigenstates that move from below $E+\delta E$ to above $E+\delta E$ is, of course, given by $N_{Y}\left(E+\delta E\right)$. The difference

$N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,$

is thus the net contribution to the increase in $\Omega$. Note that if Y dx is larger than $\delta E$ there will be the energy eigenstates that move from below E to above $E+\delta E$. They are counted in both $N_{Y}\left(E\right)$ and $N_{Y}\left(E+\delta E\right)$, therefore the above expression is also valid in that case.

Expressing the above expression as a derivative w.r.t. E and summing over Y yields the expression:

$\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,$

The logarithmic derivative of $\Omega$ w.r.t. x is thus given by:

$\left(\frac{\partial\log\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,$

The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:

$\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,$

Combining this with

$\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,$

Gives:

$dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,$

### Derivation for systems described by the canonical ensemble

If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:

$P_{j}=\frac{\exp\left(-\frac{E_{j}}{k T}\right)}{Z}$

Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:

$S = -k\sum_{j}P_{j}\log\left(P_{j}\right)$

that

$dS = -k\sum_{j}\log\left(P_{j}\right)dP_{j}$

Inserting the formula for $P_{j}$ for the canonical ensemble in here gives:

$dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}$

### General derivation from unitarity of quantum mechanics

The time development operator in quantum theory is unitary, because the Hamiltonian is hermitian. Consequently the transition probability matrix is doubly stochastic, which implies the Second Law of Thermodynamics.[13][14] This derivation is quite general, based on the Shannon entropy, and does not require any assumptions beyond unitarity, which is universally accepted. It is a consequence of the irreversibility or singular nature of the general transition matrix.

## Non-equilibrium states

Statistically it is possible for a system to achieve moments of non-equilibrium. In such statistically unlikely events where hot particles "steal" the energy of cold particles enough that the cold side gets colder and the hot side gets hotter, for an instant. Such events have been observed at a small enough scale where the likelihood of such a thing happening is significant.[15] The physics involved in such an event is described by the fluctuation theorem.

## Controversies

### Maxwell's demon

James Clerk Maxwell

James Clerk Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other. Observing the molecules on both sides, an imaginary demon guards a trapdoor between the two parts. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.

One of the most famous responses to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy. But later exceptions were found.

Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behavior of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict; hence the paradox.

One approach to handling Loschmidt's paradox is the fluctuation theorem, proved by Denis Evans and Debra Searles, which gives a numerical estimate of the probability that a system away from equilibrium will have a certain change in entropy over a certain amount of time. The theorem is proved with the exact time reversible dynamical equations of motion and the Axiom of Causality. The fluctuation theorem is proved utilizing the fact that dynamics is time reversible. Quantitative predictions of this theorem have been confirmed in laboratory experiments at the Australian National University conducted by Edith M. Sevick et al. using optical tweezers apparatus.

In statistical mechanics, a simple derivation of the entropy of an ideal gas based on the Boltzmann distribution yields an expression for the entropy which is not extensive (is not proportional to the amount of gas in question). This leads to an apparent paradox known as the Gibbs paradox, allowing, for instance, the entropy of closed systems to decrease, violating the second law of thermodynamics.

The paradox is averted by recognizing that the identity of the particles does not influence the entropy. In the conventional explanation, this is associated with an indistinguishability of the particles associated with quantum mechanics. However, a growing number of papers now take the perspective that it is merely the definition of entropy that is changed to ignore particle permutation (and thereby avert the paradox). The resulting equation for the entropy (of a classical ideal gas) is extensive, and is known as the Sackur-Tetrode equation.

### Poincaré recurrence theorem

The Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence, which is of the order of $\sim \exp\left(S/k\right)$.[16] The result applies to physical systems in which energy is conserved. The Recurrence theorem apparently contradicts the Second law of thermodynamics, which says that large dynamical systems evolve irreversibly towards the state with higher entropy, so that if one starts with a low-entropy state, the system will never return to it. There are many possible ways to resolve this paradox, but none of them is universally accepted[citation needed]. The most reasonable argument is that for typical thermodynamical systems the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, it is infinite.

### Heat death of the universe

According to the second law, the entropy of any isolated system, such as the entire universe, never decreases. If the entropy of the universe has a maximum upper bound then when this bound is reached the universe has no thermodynamic free energy to sustain motion or life, that is, the heat death is reached.

## Quotes

The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.
—Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)
The tendency for entropy to increase in isolated systems is expressed in the second law of thermodynamics — perhaps the most pessimistic and amoral formulation in all human thought.
There are almost as many formulations of the second law as there have been discussions of it.
—Philosopher / Physicist P.W. Bridgman, (1941)

## Notes

1. ^ Translated from German original: Es gibt keine Zustandsänderung, deren einziges Ergebnis die Übertragung von Wärme von einem Körper niederer auf einen Körper höherer Temperatur ist.

## References

1. ^ A. Bejan, (2006). 'Advanced Engineering Thermodynamics', Wiley. ISBN 0-471-67763-9
2. ^ J. S. Dugdale (1996, 1998). Entropy and its Physical Meaning. Tayler & Francis. p. 13. ISBN [[Special:BookSources/974-8405-69-0|974-8405-69-0]]. "This law is the basis of temperature."
3. ^ a b E. H. Lieb, J. Yngvason (1999). "The Physics and Mathematics of the Second Law of Thermodynamics". Physics Reports 310: 1–96. arXiv:cond-mat/9708200. Bibcode 1999PhR...310....1L. DOI:10.1016/S0370-1573(98)00082-9.
4. ^ a b c "Concept and Statements of the Second Law". web.mit.edu. Retrieved 2010-10-07.
5. ^ C. Caratheodory (1909). "Untersuchungen über die Grundlagen der Thermodynamik". Mathematische Annalen 67: 363. "Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können."
6. ^ H.A. Buchdahl (1966). The Concepts of Classical Thermodynamics. Cambridge University Press. p. 68.
7. ^
8. ^ Gemmer, Jochen; Otte, Alexander; Mahler, Günter (2001). "Quantum Approach to a Derivation of the Second Law of Thermodynamics". Physical Review Letters 86 (10): 1927–1930. arXiv:quant-ph/0101140. Bibcode 2001PhRvL..86.1927G. DOI:10.1103/PhysRevLett.86.1927. PMID 11289822
9. ^ Carroll; Jennifer Chen (2335). "Does Inflation Provide Natural Initial Conditions for the Universe?". Gen.Rel.Grav. () ; International Journal of Modern Physics D D14 () -2340 37 (2005): 1671–1674. arXiv:gr-qc/0505037. Bibcode 2005GReGr..37.1671C. DOI:10.1007/s10714-005-0148-2.
10. ^ Wald, R (2006). "The arrow of time and the initial conditions of the universe". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 37 (3): 394–398. DOI:10.1016/j.shpsb.2006.03.005.
11. ^ Stoner (2000). "Inquiries into the Nature of Free Energy and Entropy in Respect to Biochemical Thermodynamics". Entropy 2 (3): 106–141. arXiv:physics/0004055. Bibcode 2000Entrp...2..106S. DOI:10.3390/e2030106.
12. ^ Clausius, R. (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies. London: John van Voorst, 1 Paternoster Row. MDCCCLXVII.
13. ^ Hugh Everett, "Theory of the Universal Wavefunction", Thesis, Princeton University, (1956, 1973), Appendix I, pp 121 ff, in particular equation (4.4) at the top of page 127, and the statement on page 29 that "it is known that the [Shannon] entropy [...] is a monotone increasing function of the time."
14. ^ Bryce Seligman DeWitt, R. Neill Graham, eds, The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press (1973), ISBN 0-691-08131-X Contains Everett's thesis: The Theory of the Universal Wavefunction, pp 3–140.
15. ^ Wang, G.; Sevick, E.; Mittag, Emil; Searles, Debra; Evans, Denis (2002). "Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales". Physical Review Letters 89 (5). Bibcode 2002PhRvL..89e0601W. DOI:10.1103/PhysRevLett.89.050601.
16. ^ L. Dyson, J. Lindesay and L. Susskind, Is There Really a de Sitter/CFT Duality, JHEP 0208, 45 (2002)

• Goldstein, Martin, and Inge F., 1993. The Refrigerator and the Universe. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry. ISBN 978-0-674-75324-2
• Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. Maxwell's Demon 2 : Entropy, classical and quantum information, computing. Bristol UK; Philadelphia PA: Institute of Physics. ISBN 978-0-585-49237-7
• Halliwell, J.J. (1994). Physical Origins of Time Asymmetry. Cambridge. ISBN 0-521-56837-4. (technical).
• Iftime (2010). "Complex aspects of gravity". arXiv:1001.4571 [physics.gen-ph].
• Carnot, Sadi; Thurston, Robert Henry (editor and translator) (1890). Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power. New York: J. Wiley & Sons.  (full text of 1897 ed.)) (html)

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