Second law of thermodynamics

The historical origin[27] of the second law of thermodynamics was in Sadi Carnot's theoretical analysis of the flow of heat in steam engines (1824). The centerpiece of that analysis, now known as a Carnot engine, is an ideal heat engine fictively operated in the limiting mode of extreme slowness known as quasi-static, so that the heat and work transfers are between subsystems that are always in their own internal states of thermodynamic equilibrium. It represents the theoretical maximum efficiency of a heat engine operating between any two given thermal or heat reservoirs at different temperatures. Carnot's principle was recognized by Carnot at a time when the caloric theory represented the dominant understanding of the nature of heat, before the recognition of the first law of thermodynamics, and before the mathematical expression of the concept of entropy. Interpreted in the light of the first law, Carnot's analysis is physically equivalent to the second law of thermodynamics, and remains valid today. Some samples from his book are:

...wherever there exists a difference of temperature, motive power can be produced.[28]

The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...[29]

The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.[30]

In modern terms, Carnot's principle may be stated more precisely:

The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures.[31][32][33][34][35][36]

The German scientist Rudolf Clausius laid the foundation for the second law of thermodynamics in 1850 by examining the relation between heat transfer and work.[37] His formulation of the second law, which was published in German in 1854, is known as the Clausius statement:

Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.[38]

The statement by Clausius uses the concept of 'passage of heat'. As is usual in thermodynamic discussions, this means 'net transfer of energy as heat', and does not refer to contributory transfers one way and the other.

Heat cannot spontaneously 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 is transferred from cold to hot, but only when forced by an external agent, the refrigeration system.

Lord Kelvin expressed the second law in several wordings.

It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature.

It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.[39]

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 (The drained heat is fully converted to work.) in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the right figure. The efficiency of a normal heat engine is η and so the efficiency of the reversed heat engine is 1/η. The net and sole effect of the combined pair of engines is to transfer heat ${\textstyle \Delta Q=Q\left({\frac {1}{\eta }}-1\right)}$ from the cooler reservoir to the hotter one, which violates the Clausius statement. This is a consequence of the first law of thermodynamics, as for the total system's energy to remain the same; ${\textstyle {\text{Input}}+{\text{Output}}=0\implies (Q+Q_{c})-{\frac {Q}{\eta }}=0}$, so therefore ${\textstyle Q_{c}=Q\left({\frac {1}{\eta }}-1\right)}$, where (1) the sign convention of heat is used in which heat entering into (leaving from) an engine is positive (negative) and (2) ${\displaystyle {\frac {Q}{\eta }}}$ is obtained by the definition of efficiency of the engine when the engine operation is not reversed. 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.

Planck offered the following proposition as derived directly from experience. This is sometimes regarded as his statement of the second law, but he regarded it as a starting point for the derivation of the second law.

It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.[40][41]

It is almost customary in textbooks to speak of the "Kelvin–Planck statement" of the law, as for example in the text by ter Haar and Wergeland.[42] This version, also known as the heat engine statement, of the second law states that

It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.[2]

Planck stated the second law as follows.

Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.[43][44][45]

Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.

... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.[46]

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:[47]

In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.[48]

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. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, ${\displaystyle \delta Q=TdS}$.[49]

Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.[17][50][51][52]

In 1926, Max Planck wrote an important paper on the basics of thermodynamics.[51][53] He indicated the principle

The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.[17][50]

This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work."[54] Planck wrote: "The production of heat by friction is irreversible."[55][56]

Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above.[57] It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.

A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:

... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.[58]

Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.

The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).[59]

Before 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'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 realize 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.

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

${\displaystyle \oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0.}$

The equality holds in the reversible case[60] and the strict inequality holds in the irreversible case, with T_surr as the temperature of the heat bath (surroundings) here. 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 from state functionality.

For an arbitrary heat engine, the efficiency is:

${\displaystyle \eta ={\frac {|W_{n}|}{q_{H}}}={\frac {q_{H}+q_{C}}{q_{H}}}=1-{\frac {|q_{C}|}{|q_{H}|}}}$

(1)

where W_n is the net work done per cycle, q_H > 0 is the heat added from a hot reservoir, and q_C = - |q_C| < 0 [61] is waste heat given off to a cold reservoir. Thus the efficiency depends only on |q_C|/|q_H|.

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 T_H and T_C must have the same efficiency, that is to say, the efficiency is a function of temperatures only:

${\displaystyle {\frac {|q_{C}|}{|q_{H}|}}=f(T_{H},T_{C}).}$

(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 and T₃. This can only be the case if

${\displaystyle f(T_{1},T_{3})={\frac {q_{3}}{q_{1}}}={\frac {|q_{2}||q_{3}|}{|q_{1}||q_{2}|}}=f(T_{1},T_{2})f(T_{2},T_{3}).}$

Now consider the case where ${\displaystyle T_{1}}$ is a fixed reference temperature: the temperature of the triple point of water. Then for any T₂ and T₃,

${\displaystyle f(T_{2},T_{3})={\frac {f(T_{1},T_{3})}{f(T_{1},T_{2})}}={\frac {273.16{\text{ K}}\cdot f(T_{1},T_{3})}{273.16{\text{ K}}\cdot f(T_{1},T_{2})}}.}$

Therefore, if thermodynamic temperature is defined by

${\displaystyle T=273.16{\text{ K}}\cdot f(T_{1},T)}$

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

${\displaystyle f(T_{2},T_{3})={\frac {T_{3}}{T_{2}}},}$

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

According to the Clausius equality, for a reversible process

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

That means the line integral ${\displaystyle \int _{L}{\frac {\delta Q}{T}}}$ is path independent for reversible processes.

So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies

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

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 states 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 the Clausius inequality on this loop, with T_surr as the temperature of the surroundings,

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

Thus,

${\displaystyle \Delta S\geq \int {\frac {\delta Q}{T_{surr}}}}$

where the equality holds if the transformation is reversible.

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

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 T_R and pressure P_R  – so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T_R; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P_R.

Whatever changes to dS and dS_R occur in the entropies of the sub-system and the surroundings individually, according to the second law the entropy S_tot of the isolated total system must not decrease:

${\displaystyle dS_{\mathrm {tot} }=dS+dS_{R}\geq 0}$

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 Σμ_iRN_i, so that:

${\displaystyle dU=\delta q-\delta w+d\left(\sum \mu _{iR}N_{i}\right)}$

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

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

${\displaystyle \delta q=T_{R}(-dS_{R})\leq T_{R}dS}$

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

${\displaystyle \delta w\leq -dU+T_{R}dS+\sum \mu _{iR}dN_{i}}$

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

${\displaystyle \delta w_{u}\leq -d\left(U-T_{R}S+p_{R}V-\sum \mu _{iR}N_{i}\right)}$

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

${\displaystyle E=U-T_{R}S+p_{R}V-\sum \mu _{iR}N_{i}}$

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,

${\displaystyle dE+\delta w_{u}\leq 0}$

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 E for an irreversible process and no change for a reversible process.

${\displaystyle dS_{tot}\geq 0}$ is equivalent to ${\displaystyle dE+\delta w_{u}\leq 0}$

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.

For a spontaneous chemical process in a closed system at constant temperature and pressure without non-PV work, the Clausius inequality ΔS > Q/T_surr transforms into a condition for the change in Gibbs free energy

${\displaystyle \Delta G<0}$

or dG < 0. For a similar process at constant temperature and volume, the change in Helmholtz free energy must be negative, ${\displaystyle \Delta A<0}$. Thus, a negative value of the change in free energy (G or A) is a necessary condition for a process to be spontaneous. This is the most useful form of the second law of thermodynamics in chemistry, where free-energy changes can be calculated from tabulated enthalpies of formation and standard molar entropies of reactants and products.[18][14] The chemical equilibrium condition at constant T and p without electrical work is dG = 0.

Rudolf Clausius

In 1865, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:[65]

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

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. Because of the looseness of its language, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This is not true; this statement is only a simplified version of a more extended and precise description.

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

${\displaystyle {\frac {dS}{dt}}\geq 0}$

where

S is the entropy of the system and

t is time.

The equality sign applies after equilibration. An alternative way of formulating of the second law for isolated systems is:

${\displaystyle {\frac {dS}{dt}}={\dot {S}}_{i}}$ with ${\displaystyle {\dot {S}}_{i}\geq 0}$

with ${\displaystyle {\dot {S}}_{i}}$ 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 ${\displaystyle T_{a}}$ it gives the so-called dissipated energy ${\displaystyle P_{diss}=T_{a}{\dot {S}}_{i}}$.

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

${\displaystyle {\frac {dS}{dt}}={\frac {\dot {Q}}{T}}+{\dot {S}}_{i}}$ with ${\displaystyle {\dot {S}}_{i}\geq 0}$

Here

${\displaystyle {\dot {Q}}}$ is the heat flow into the system

${\displaystyle T}$ is the temperature at the point where the heat enters the system.

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. 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):

${\displaystyle {\frac {dS}{dt}}={\frac {\dot {Q}}{T}}+{\dot {S}}+{\dot {S}}_{i}}$ with ${\displaystyle {\dot {S}}_{i}\geq 0}$

Here ${\displaystyle {\dot {S}}}$ 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.

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

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

where the temperature is defined as:

${\displaystyle {\frac {1}{k_{\mathrm {B} }T}}\equiv \beta \equiv {\frac {d\ln \left[\Omega \left(E\right)\right]}{dE}}}$

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 ${\displaystyle Xdx}$ is the work performed by the system if x is increased by an amount dx. For example, if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate ${\displaystyle E_{r}}$ is given by:

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

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

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

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

${\displaystyle \Omega \left(E\right)=\sum _{Y}\Omega _{Y}\left(E\right)\,}$

The average defining the generalized force can now be written:

${\displaystyle X=-{\frac {1}{\Omega \left(E\right)}}\sum _{Y}Y\Omega _{Y}\left(E\right)\,}$

We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then ${\displaystyle \Omega \left(E\right)}$ will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between ${\displaystyle E}$ and ${\displaystyle E+\delta E}$. Let's focus again on the energy eigenstates for which ${\textstyle {\frac {dE_{r}}{dx}}}$ lies within the range between ${\displaystyle Y}$ and ${\displaystyle Y+\delta Y}$. 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

${\displaystyle N_{Y}\left(E\right)={\frac {\Omega _{Y}\left(E\right)}{\delta E}}Ydx\,}$

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

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

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

Expressing the above expression as a derivative with respect to E and summing over Y yields the expression:

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

The logarithmic derivative of ${\displaystyle \Omega }$ with respect to x is thus given by:

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

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 vanish in the thermodynamic limit. We have thus found that:

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

Combining this with

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

gives:

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

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:

${\displaystyle P_{j}={\frac {\exp \left(-{\frac {E_{j}}{k_{\mathrm {B} }T}}\right)}{Z}}}$

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:

${\displaystyle S=-k_{\mathrm {B} }\sum _{j}P_{j}\ln \left(P_{j}\right)}$

that

${\displaystyle dS=-k_{\mathrm {B} }\sum _{j}\ln \left(P_{j}\right)dP_{j}}$

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

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

As elaborated above, it is thought that the second law of thermodynamics is a result of the very low-entropy initial conditions at the Big Bang. From a statistical point of view, these were very special conditions. On the other hand, they were quite simple, as the universe - or at least the part thereof from which the observable universe developed - seems to have been extremely uniform.[71]

This may seem somewhat paradoxical, since in many physical systems uniform conditions (e.g. mixed rather than separated gases) have high entropy. The paradox is solved once realizing that gravitational systems have negative heat capacity, so that when gravity is important, uniform conditions (e.g. gas of uniform density) in fact have lower entropy compared to non-uniform ones (e.g. black holes in empty space).[72] Yet another approach is that the universe had high (or even maximal) entropy given its size, but as the universe grew it rapidly came out of thermodynamic equilibrium, its entropy only slightly increased compared to the increase in maximal possible entropy, and thus it has arrived at a very low entropy when compared to the much larger possible maximum given its later size.[73]

As for the reason why initial conditions were such, one suggestion is that cosmological inflation was enough to wipe off non-smoothness, while another is that the universe was created spontaneously where the mechanism of creation implies low-entropy initial conditions.[74]

Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from the time-symmetric dynamics that describe the microscopic evolution of a macroscopic system.

In the opinion of Schrödinger, "It is now quite obvious in what manner you have to reformulate the law of entropy – or for that matter, all other irreversible statements – so that they be capable of being derived from reversible models. You must not speak of one isolated system but at least of two, which you may for the moment consider isolated from the rest of the world, but not always from each other."[88] The two systems are isolated from each other by the wall, until it is removed by the thermodynamic operation, as envisaged by the law. The thermodynamic operation is externally imposed, not subject to the reversible microscopic dynamical laws that govern the constituents of the systems. It is the cause of the irreversibility. The statement of the law in this present article complies with Schrödinger's advice. The cause–effect relation is logically prior to the second law, not derived from it.

The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.

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, separated by a wall. Observing the molecules on both sides, an imaginary demon guards a microscopic trapdoor in the wall. 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 response 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.

Maxwell's 'demon' repeatedly alters the permeability of the wall between A and B. It is therefore performing thermodynamic operations on a microscopic scale, not just observing ordinary spontaneous or natural macroscopic thermodynamic processes.