Thermodynamic entropy of a many body energy eigenstate

It is argued that a typical many body energy eigenstate has a well defined thermodynamic entropy and that individual eigenstates possess thermodynamic characteristics analogous to those of generic isolated systems. We examine large systems with eigenstate energies equivalent to finite temperatures. When quasi-static evolution of a system is adiabatic (in the quantum mechanical sense), two coupled subsystems can transfer heat from one subsystem to another yet remain in an energy eigenstate. To explicitly construct the entropy from the wave function, degrees of freedom are divided into two unequal parts. It is argued that the entanglement entropy between these two subsystems is the thermodynamic entropy per degree of freedom for the smaller subsystem. This is done by tracing over the larger subsystem to obtain a density matrix, and calculating the diagonal and off-diagonal contributions to the entanglement entropy.


Introduction
The main question investigated here is if it is possible to define a thermodynamic entropy for energy eigenstates of a generic many body system, and if so, how this can be characterized by examining the wave function in such a state. The thermodynamic entropy S, is measurable experimentally by changing parameters such as temperature T , and calculating small changes in heat dQ via the relation dS = dQ/T. (1) By incrementally measuring dS relative to a reference state, say at low temperature, the entropy can be calculated. The entropy is important because it is only a function of thermodynamic state variables, not a system's history, and therefore it should be, in principle, calculable without changing external parameters. Classically, the Boltzmann hypothesis, that the entropy is the logarithm of the number of states dynamically accessible to a system, appears to be correct for generic systems in strict thermal equilibrium [1]. This gives a means to determine system's entropy with external parameters held constant. In quantum mechanics, our understanding of entropy is not as well developed. A theoretical calculation of the entropy is possible via the free energy, using the canonical ensemble. An ensemble of systems are summed over, each one with a different energy. Although this appears to work in practice, the reason why this canonical entropy formula works for a pure state is not clear. The calculation assumes the system is in a mixed state, with microcanonical or canonical weights. However one expects that thermodynamics should apply to pure states, so it is far from clear that the canonical entropy calculation applies to that case. The Von Neumann entropy for a system with a density matrix ρ is S V N (ρ) = − tr ρ ln ρ, and is zero for a system in a pure state, although it gives the usual result for the entropy using a canonical mixed state. As a result, S V N is clearly not a candidate for the thermodynamic entropy of a pure state.
On the other hand, if we consider a system in an energy eigenstate, it has a trivial time dependence, and it is hard to see how to associate a thermodynamic entropy to it, in any way similar to the Boltzmann hypothesis. There is no exploring of phase space with time and it is not clear from our intuition of heat, how the concept of entropy could be meaningful.
In this work I consider a single energy eigenstate of a large ergodic system, with the word "ergodic" defined in the next section. I argue that for such a state, the concept of a thermodynamic entropy is still meaningful and that such a system under extremely slow external perturbations such as a changing magnetic field, will transfer heat between different parts of a system, while remaining in an energy eigenstate in accordance with the usual laws of thermodynamics.
Because this entropy is a function of the state of the system, it should be possible to find a recipe to calculate its thermodynamic entropy from the wave function itself with no changes in external parameters. This recipe should be equal to the entropy calculated in the canonical ensemble for large systems. This lends support to the conjecture that even an energy eigenstate has thermodynamic properties seen for more generic density matrices.
To construct the entropy of an energy eigenstate, the quantity used is the "entanglement entropy". First we subdivide a system into two macroscopic systems, one labeled θ and the other labeled φ with m and n degrees of freedom respectively. One firsts defines a density matrix ρ φ = Tr θ |φ, θ φ, θ|. Then the entanglement entropy S E ≡ − tr ρ φ ln ρ φ . This has many useful properties. For example it is also equal to − tr ρ θ ln ρ θ .
The entanglement entropy plays a crucial role in quantum information theory [2] and has also been used in the study of black holes [3], and quantum phase transitions [4]. The entanglement entropy has been argued to be a unique measure of entanglement in a pure state [5] by making analogies with thermodynamics, but it's relationship to the thermodynamic entropy is still unclear.
At zero temperature, S E is non-zero but non-extensive and for a variety of systems is a power law of system size [6,7]. At finite temperature and large system size, 1+1 dimensional quantum field theories give extensive behavior.
Here we examine examine how to calculate the entanglement entropy in an energy eigenstate by means similar to those used earlier to study the question of energy eigenstate thermalization. First in Sec. 2 we discuss this problem. Applying that conjecture, we show in 3 how this implies that the notion of heat and thermodynamic entropy are meaningful for energy eigenstates. Finally in Sec. 4 we present a calculation of the entanglement entropy.
We argue that for a system in an energy eigenstate where the number of degrees of freedom go to infinity, and energies corresponding to a fixed finite temperature, the entanglement entropy becomes the thermodynamic entropy per degree of freedom. This gives an explicit formula for deducing thermodynamic properties from individual energy eigenstates.
The outline of this paper is as follows. In Sec. 2, we discuss previous work that argues that for energy eigenstates, the expectation value of a large class of observables will be equivalent to averages using the microcanonical ensemble. Using this, in Sec. 3 we explore the relation between thermodynamic entropy and its microcanonical value, for systems in energy eigenstates. In Sec. 4 we analyze the entanglement entropy, using methods similar to those of Sec. 2.

Energy Eigenstate Thermalization
Previous work by the author [8] and others [9] attempted to understand why the laws of quantum statistical mechanics work for an isolated system. The approach taken was to see what could be derived about statistical mechanics from quantum mechanics without any additional assumptions and for a particular choice of model systems. The assumption made in quantum statistical mechanics is that the average over time . . . t of some observable quantity ψ|A|ψ is equal to a microcanonical average at a total energy e that is assumed to be well defined (see below), where e j labels an energy eigenstate of the entire system and ∆(e, e j ) is a function that is sharply peaked at e = e j . For a system containing a large number of degrees of freedom and for a large class of operators A, this can be written with negligible error in terms of the canonical distribution at fixed temperature (throughout this work, units are chosen so that Boltzmann's constant is unity.) We will postpone to the end of this section how these formulas should, rather simply, be modified to take into account fluctuations in the total energy, but Eq. 2 or equivalently Eq. 3 has been enormously successful in explaining problems in almost every branch of physics. For the purposes of this paper, systems obeying this equation will be called "ergodic".
In classical mechanics, a system with a few of degrees of freedom such as Sinai billiards, have time averages given by the microcanonical distribution. A quantum mechanical treatment of the same system cannot be expected to give the microcanonical distribution. It is easy to show by counterexample, that one needs at least one more requirement; the number of degrees of the system must also be large. Indeed, if the spacing between energy levels is not small it is impossible to define a microcanonical distribution in a precise way.
Indeed, the density of states G(E) for a system with n degrees of freedom is related to the entropy, and for an extensive system G(E) = (1/e 0 ) exp(ns(E/n)). Here s is the entropy per particle and e 0 is an energy normalization. Therefore, if the energy width in the microcanonical average is δ, then the number of states being averaged over is proportional to δ exp(ns(E/n)). With fixed δ, the number of states contributing to the average diverges exponentially with n implying that fluctuations in microcanonical quantities will rapidly go to zero with increasing n as the total energy is varied.
Having a large number of degrees of freedom however, is not enough to ensure ergodicity. As a simple example, consider a perfect harmonic crystal in which case the initial choice of wave function alters the time averages of an observable. If such a system were to start with a wave function obeying Eq. 2, then shining light on it so as to couple to some modes preferentially, will now violate this microcanonical average.
Thus we must search for a mechanism that can explain how an experimental system can continue to give microcanonical averages for time averaged quantities despite atypical initial states, such as the example of light described above. The approach taken was similar to understanding how this works for an almost ideal classical gas. A genuine ideal gas has no interaction between different particles and therefore will not be ergodic, but can be made so by slight modification. For example, the particles can be given hard cores of very small diameter, which will have a negligible effect on the statistical and thermodynamic properties computed from the Gibbs distribution. However after a long enough time, the system will explore almost all of its available phase space enabling the rigorous application of the statistical mechanical formula Eq. 2. In the same spirit, suppose we start with a Hamiltonian that decouples into n separate subsystems This system is not ergodic. However the main result of the previous work [8], was to show that with negligible error in the limit of large n, it can be made ergodic by the addition of a small perturbation as described below. The model considered was where H 1 is added in the hopes of making the system ergodic. In the case of an ideal gas for example, one may want to add some interaction between the different particles, for example, take However this is extremely hard to analyze so instead of adding in these interactions explicitly, we use a random matrix model to understand the effects of a generic perturbation. If we consider the problem in the basis of energy eigenvalues of H 0 , then we model H 1 by a real symmetric matrix whose elements are chosen from a real random Gaussian ensemble, with certain physically sensible conditions on the magnitude of the elements described in the next paragraph. The use of a random matrix is sensible in this context as much work, starting with the monumental work of Wigner [10] that shows a deep connection between the physics of interacting systems and random matrices. We take the variances of these random couplings to decrease away from the matrix diagonal.
where the magnitudes of the ǫ are taken to be much less than T but much greater than the energy spacing as will be discussed in more detail below. In this paper, the line above the matrix elements denotes an average over the ensemble of random matrices. The reason that the variance of the matrix elements are taken to depend on position is explained in more detail in Appendix Appendix A, but in general, one expects asymptotically that the effect of coupling from states of different energies E 1 and E 2 , to decrease with their energy difference. The size of elements are diminished by a phase space factor for T ≪ |E 1 − E 2 | ≪ E 1 (the ground state energy of the system is set to 0). The temperature is defined by the usual prescription 1 The energy E in the above derivative can be evaluated at either E 1 or E 2 since for a large system T ≪ E. So when |E 1 − E 2 | >> T , the proportion of non-zero matrix elements is effectively zero. To simplify the model further, we consider a banded random matrix, where the width of the band increases with energy. Inside the band, all off-diagonal elements have the same variance ǫ. The precise form of the cutoff is unimportant to the conclusions but should be present on physical grounds and also prevents unphysical divergences in expectation values. The random matrix model defined above is similar to one analyzed by Wigner [11], where he considered a matrix with diagonal elements that were linearly increasing, D i = ∆i, and ∆ = 1/G(E) can be taken to be the average energy level spacing, and off-diagonal random matrix elements are banded as above with the width n b ≫ 1. Eigenvectors are random but the square of the amplitudes are well defined and are not constant. Denoting the amplitudes of the ith eigenvector by c ij , for |i − j|∆ ≪ n b ∆. In the opposite limit, the eigenvalues decay faster than an exponential. This is why finite n b prevents unphysical divergences.
As it turns out, the width of the Lorentzian δ, determines the energy width that is used when doing a microcanonical average in Eq. 2. Therefore one would like δ ≪ T , so as not to change the microcanonical result. But conversely, δ ≫ ∆ (the energy spacing). Both requirements are easily satisfied because as mentioned above, the total number of states contributing to the microcanonical average is ∼ δ exp(ns(E/n)) This means one should choose ∆ = e 0 exp(−ns(E/n)) ≪ δ ≪ T which is easily satisfied for large n, as we will take δ to be independent of n. In terms of the parameter ǫ in this model, Eq. 11 gives that ∆ ≪ ǫ ≪ √ T ∆. Because of the immense smallness of ∆ the addition of ǫ will not significantly change the partition function for this system. Instead it was shown [8] that its effect is to make infinite time averages in accord with the microcanonical distribution.
The sense that this model gives microcanonical results is as follows. One considers time averages done with one choice of random matrix. The answer will differ from that of another realization. We can compute what the variance of the average will be, averaged over all matrices in the ensemble. It was shown that this variance is proportional to ∆/δ which is exponentially small in n.
The most striking feature of this is that the equivalence to the microcanonical distribution should apply even to energy eigenstates for a large class of operators A. This is surprising because the time dependence of such wave functions is trivial and therefore does not show any chaotic time dependent behavior. Of course the spatial dependence is extremely complex and this is the reason why it can give rise to this kind of self averaging. This equivalence for energy eigenstates has been recently confirmed by ab initio numerical tests [12] and is often referred to as "Eigenstate Thermalization".
A caveat must be stated to the above claim. For an initial state with a large spread in total energies, the microcanonical distribution is not obtained because the system cannot be averaged at only one energy. Instead it can be shown [8] that time averages require an additional averaging, over the probability of finding the system at a particular energy.

Thermodynamics of Energy Eigenstates
An energy eigenstate has trivial time dependence and it is of interest to investigate whether a complete system when placed in such a state still obey thermodynamics. For an isolated system, there is no heat flow into or out of the system, so we examine a closed system composed of two macroscopic parts, A and B that are weakly interacting so that the Hamiltonian is H = H A + H B + H i . To analyze this, we will assume the thermalization of individual eigenstates [8], as argued in the last section.
Suppose only H A depends on an external parameter, x. For example, x could be an external magnetic field or the position of a piston. Because H B does not depend on the external parameter, one could regard B as a heat bath for subsystem A, though this would only be a good analogy when subsystem B was much larger then A. For example, A could be a gas cylinder with a movable piston at position x, and B could be a heat bath in contact with A. We can analyze how energy gets transfered between A and B as a result of changing x infinitesimally and quasistatically from x to x + dx. By quasistatic, we mean adiabatic in the quantum mechanical sense. We are also considering a completely isolated system, so that it is adiabatic in the thermodynamic sense. However the subsystems can transfer energy between each other.
Because we are only interested in systems that are not integrable (and in fact "ergodic" in the sense of the word given above), we expect that energy level repulsion will prevent any level crossing during a change in x, so it is possible, in principle, to vary a parameter and stay in an energy eigenstate.
Since in an energy eigenstate E depends on x, by the Guttinger-Feynman-Hellman theorem [13] Because of our assumption of individual eigenstate thermalization, where the subscript m under the averages denote a microcanonical average. The right hand side implies that the energy derivative is the same as for a generic mixed state (with the usual assumption of a sharply peaked energy distribution). Because for a large system the microcanonical average is equivalent to a canonical average, and the average only involves subsystem A, we take the average over only subsystem A using say, a microcanonical ensemble for A. Therefore it is possible to relate this to the microcanonical (of canonical) definition of entropy in A similar to standard procedures, see for example Reif [14]. First, the work dW done by the system when x is changed quasistatically to x + dx is Note that the rate must be slow enough for the system to remain almost entirely in an energy eigenstate. The work done, say for example by moving a piston, involves a change in energy of both parts A and B. Although the piston is part of system A, a flow of energy from B to A can also occur, and such a flow will contribute to the total amount of work done in changing x. This is familiar in the common thermodynamics example of the expansion of a gas under adiabatic conditions, or under constant temperature conditions. In the latter case, heat flows from a heat bath into the gas contributing to the work done. Therefore the work done involves the total change in the energy of A and B although the piston is only attached to A. Second, statistical mechanical entropy of system A, S A is defined as S A = ln Ω A , where Ω A is the number of states in the energy window being considered. Then as shown by Reif [14] ∂S The microcanonical average is being taken at an energy E A = ψ|H A |ψ for only subsystem A. Appendix Appendix B gives a simple derivation of this. Then one writes the differential Note from Eq. 13 and 15 The last equality involves dE A + dW which is usual definition of the total heat dQ absorbed by A. Hence Therefore the ensemble definition of the entropy, is related to the flow of energy dQ between sub-systems A and B by the usual thermodynamic relation, Eq. 18. For thermodynamic purposes, the energy flow dQ is completely equivalent to heat yet it is seen for a system in an energy eigenstate. A change in an external parameter acting on one part of a system in an energy eigenstate will cause entropy to redistribute itself across subsystems.
Although the above does not show that there is an entropy associated with the complete system in an energy eigenstate, it shows that entropy changes of sub-systems can be induced by a slowly varying external parameter. The interaction between the sub-systems could also be slowly switched off, leading to two systems both in energy eigenstates. By first slowly changing x and then slowly switching off their interaction, one can compare the difference in entropies for different final values of x. This is similar to the procedure used in determining entropy using reference states. Therefore relative entropies between different energy eigenstates can be calculated. This suggests that the absolute entropy for a system in an energy eigenstate is a meaningful concept.
In the next section, we argue that this thermodynamic entropy associated with an energy eigenstate and be calculated by means of the entanglement entropy.

The Model
As discussed earlier in Sec. 1, we consider a system in an eigenstate energy E and with n tot degrees of freedom. We subdivide it into two macroscopic systems, one φ and the other θ with n and m degrees of freedom respectively. We will consider n large but m ≫ n. If both sub-systems were uncoupled, we can diagonalize each of them into energy eigenstates forming a complete set of |φ i and |θ j . The wave function for the complete system can then be written as The summation is over all states. Consider the case where the coupling Hamiltonian H 1 between m and n is weak. Then to first order in perturbation theory an energy eigenstate of the uncoupled system |φ 0 |θ 0 is altered by H 1 as follows As we argued above, θ j | φ i |H 1 |φ 0 |θ 0 will become exponentially small when the energy difference between the bra and ket states is much greater than T . Higher order terms in the perturbation series also have this property. Therefore applying a perturbation that couples n and m will only allow a coupling if their energies differ by a microscopic energy of order T . Because coefficients C ij are identical to those in Eq. 20, the only ones that need be considered are those where the energy of state |φ i |θ j is almost constant.
Although I expect the arguments here to hold more generally, one can make the model more precise by adopting an approach similar to that used in the Sec. 3. We choose the energy eigenvectors as a basis for |ψ before coupling H 1 is turned on. Therefore the energy in an eigenstate |φ |θ is the sum of the energy of φ plus the energy of θ. When the interaction H 1 is switched on then the effects of this coupling can be modeled as was done in Sec. 3 as a banded random matrix. In this case, C ij C kl , averaged over different realizations of H 1 , are zero for i = k or j = l.
The argument that we present now is analogous to the standard argument of energy exchange between two subsystems [1]. Out of states |φ i |θ j that will contribute to |ψ , there will be ones where the energy e θ of subsystem θ is high and e φ is low, and vice versa. Because the density of states of each subsystem increases extremely rapidly with energy, there will be a very sharp peak in the number of states that contribute as a function of e θ . This means that the two subsystems will be at the same "temperature" and the standard deviation in energy of one subsystem ∆E is ∝ Cnm/(n + m)T where C is the specific heat per degree of freedom. This is much greater than the microscopic thermal energy T but much less than the total energy of |φ .

Entanglement Entropy Calculation
The density matrix of the φ, in the |φ |θ basis, tracing over θ can be represented as First we will examine the diagonal portion of this density matrix, ρ ii . By completeness, tr(ρ) = ψ|ψ = 1. By integrating over a subsystem m, ρ ii will be close to zero outside of a window that depends on the size of m. To estimate the size of this window, we note that total energy of the isolated system is conserved. First consider ρ ii averaged over an ensemble of C ′ s, (or random matrices) The number of states effectively contributing, M, is finite, because as we just argued, conservation of energy and the very sharp peak in the density of states as a function of e θ , imply that only states of θ with an energy within a few windows of ∆E, will contribute to this sum. Therefore M ∝ ∆E exp(ms(E/n tot )), s being the microcanonical entropy per degree of freedom. In Eq. 22, we expect that |C ij | 2 to be a smooth function of the j's in analogy to Eq. 10. Because i ρ ii = 1, we can estimate the entropy as follows.
The number of terms contributing to this sum is N = ∆EG(E) where G(E) is the density of states of subsystem φ and is ∝ exp(ns(E/n tot ). We take the terms in this window to be constant so that ρ ii = 1/N. To get our initial estimate, we will assume that all off-diagonal components are negligible (but we will do a better job below.) Therefore this entropy estimate is then − i ρ ii ln ρ ii , which is then N(1/N) ln(N) But N ∝ √ n exp(ns(E/n tot )). For large N, this becomes nS(E/n tot ) + O(ln n). For large n, this becomes precisely the entropy of the subsystem φ.
The reason why such a crude approximation to the density matrix gives the correct answer is the same reason as it works with many statistical mechanical calculations. Only the peak value of ρ ii matters after taking a logarithm. Therefore one could instead have taken a non-flat distribution for ρ ii and this would not have altered the dominant term. However by pre-averaging ρ, we ignored fluctuations which will be sizable if m is small. To estimate these we expand ρ ii = ρ ii + δρ ii . Then we average to find the size of the fluctuations. Expanding ρ ii ln ρ ii in δρ ii and averaging, the first order terms vanishes, leaving a correction δρ ii 2 /(2ρ ii ).
With uncorrelated C ij 's, δρ ii 2 = var(ρ ii ) ∼ Mvar(|C ij | 2 ). We also expect that the C ij 's will be close to Gaussian, which means that var( Therefore we expect corrections to the diagonal elements of the density matrix will be exponentially small in the system size m.
We will now turn to a calculation of the contribution of the entanglement entropy due to off-diagonal matrix elements. We will first estimate the size of the off-diagonal portions of the density matrix, and to do this we will again assume as with a random matrix model, that as above, the C ij have random phase and be uncorrelated. Averaging over random realizations of the C's gives and estimate for the variance of ρ ij For i = j, Because this is much smaller than the order of ρ ∼ O(1/N), it makes sense to treat the off diagonal elements perturbatively. We expand the density matrix where D ij is the diagonal part of the density matrix and ǫ ij is the remaining off-diagonal terms, with ǫ ii = 0. The entropy is S(ρ) = − tr(ρ ln ρ)). We expand the entropy around ρ = D to second order in ǫ. This is calculated in Appendix Appendix C and gives The function F (x, y), defined in Eq. C.13, is symmetric in its arguments and is peaked along the line x = y, where it has the value F (x, x) = 1/x. As x and y go to zero, the summand in Eq. 26 is well behaved as can be checked as follows.
By Schwartz's inequality Eq. 21 implies Therefore |F (ρ nn , ρ mm )Re(ǫ 2 nm )| ≤ |F (ρ nn , ρ mm )ρ mm ρ nn | And it is easily seen that the function F (x, y)xy is well behaved for small x and y. Now we are in a position to estimate the order of the off-diagonal contribution S 2 to the total entanglement entropy S using Eq. 26. With N 2 terms in the sum, ρ 2 nm estimated using Eq. 24, and F contributing O(ln ρ ii /ρ ii ) = O(ln N/(1/N)) = O (N ln N), this gives The diagonal term, which is also the thermodynamic entropy is ln N. Therefore for N ≪ M the off-diagonal contribution to the entropy is negligible and the entropy is given by the ensemble result. Also note that for a homogeneous system in order for S 2 to be small compared to the canonical result, one does not require n ≪ m. Because ln(N/M) ∝ n − m one instead requires that m − n is large. For a macroscopic system, m − n can be made very large while (m − n)/m can be very small. This suggests that in the limit of large n, the entanglement entropy will be equivalent to the canonical entropy for n/m < 1.
Because the entanglement entropy of φ is identical [15] to that of θ, for n/m > 1, the entropy obtained from the entanglement entropy becomes that of system θ.

Discussion
There are many definitions of th entropy, and it is often couched in terms of the lack of information about a system. Classically, the relationship between information and the system's state is straightforward. If the microscopic state of a system is completely characterized, this means that we have all possible information about it. But a gas applies the same time-averaged pressure to a piston irrespective of the experimenter's state of ignorance. The entropy used for thermodynamic purposes is not dependent on our knowledge of the system, so that derivatives of the entropy with respect to parameters such as volume via Eq. 15, give us experimentally measurable quantities such as the pressure. This is why in this work I have concentrated on understanding the thermodynamic entropy. This entropy is calculable through standard means, such as the canonical ensemble.
However the situation becomes more murky when considering quantum mechanics. The process of obtaining a precise state involves measurement, which couples the system of interest to another system. This changes its state. Therefore the act of measuring a system's energy precisely, so as to put it into an energy eigenstate, might then effect its thermodynamic entropy.
However for finite temperature systems, the above results suggest that a generic many body system in an energy eigenstate has a well defined thermodynamic entropy that is calculable using the usual statistical mechanical methods, for example, through the canonical ensemble. From the above argument, this is not obvious and it is virtually impossible to test this experimentally on a large system because the separation between energy levels is exponentially small in the number of degrees of freedom.
If a system starts out in some generic pure state with many different components in different energy eigenstates, and one measures the energy precisely, this puts it into an energy eigenstate. One can construct an argument suggesting that contrary to the claims here, the entropy after such a measurement would be greatly effected. If the system starts out with an energy spread over some width, for example corresponding to the wavelength of a box size, it is the sum of an exponentially large number N ∝ exp(S) energy eigenstates |e i so that |ψ >= i a i |e i with some coefficients a i . If the energy is measured to sufficient accuracy so as to put into a single energy eigenstate, then this has reduced the number of coefficients in this sum down to one. The energy measurement outputs a number of great accuracy describable by a minimum of log 2 (N) bits. This is a large number that is extensive in the size of the system. This much information being produced, and the collapse of the wave function to a single energy eigenstate, might give one reason to believe that the thermodynamic entropy has been reduced by an extensive amount.
However there are two problems with this argument. First as we argued above for the classical regime, the act of measurement process does not alter the thermodynamic entropy. Second, a measurement by itself does not necessarily cost any energy. As shown by Bennett [16], it does not cost energy to find the state of a two-state system, if this is done sufficiently slowly. If the apparatus was originally in a known standard state, it will find itself in a different state after the measurement process, and that state will depend on the outcome of the measurement. In order for further measurements to take place, the apparatus needs to be reset which means that the phase space of the measurement apparatus must be contracted, which will cost an energy k B T /2.
One can also see that putting the system in contact with a small system of order just a few degrees of freedom at the same temperature will immediately destroy the energy eigenstate, returning it to a wave function with many energy components. Therefore, it is not plausible that measuring the system to this accuracy could reduce its entropy by a macroscopic amount. As far as interaction with other systems, it is expected to behave as a more generic pure state at the same temperature.
I have calculated the entanglement entropy of an energy eigenstate with an energy equivalent to a system at finite temperature, to leading order in the size of the system. For notational simplicity, I have occasionally assumed that the system is homogeneous, however the results should apply for non-homogeneous systems. The complete entropy is obtained by dividing the system several ways into a larger and a smaller subsystem and calculating the entanglement entropy between them.
There will be corrections to this prescription that become small for large system size. Exact results for zero temperature systems is an indication of the presence of sub-extensive terms that have many interesting applications [7,6]. The methods used here do not easily give non-extensive corrections and consequently do not give these zero temperature results.
The result that the entanglement entropy of a finite temperature system is equal to the thermodynamic entropy can be shown rigorously to be the case in special cases [17,18,19]. This was shown to be the case for 1+1 dimensional conformal field theories in a finite temperature mixed state ensemble, where the system of interest is connected to an infinite system. It is not surprising that in this limit, one obtains the usual entropy as a canonical ensemble has been used which acts, in effect, as a heat bath. In this paper, we analyzed the case where the system is in an energy eigenstate and reached the same conclusion concerning the entanglement entropy. In this case, there is no heat bath, only the coupling of the subsystem of interest to the rest of the system. It is not obvious that such a coupling should be enough to result in the answer one obtains for a mixed state. The model that we used was similar to that of previous work [8], and assumes that the coupling between the two parts of the system φ and θ is sufficiently chaotic to be describable by random matrices. This is certainly an approximation and for short range forces, the interactions between the two should be taking place only on the interface between the two subsystems. However I conjecture that the result is robust enough to apply so such cases. Numerical work should in principle, be able to test the arguments presented here.
We also looked at the effects of external forces that are slowly applied to an energy eigenstate. We argued that such states are no different thermodynamically than for generic pure states. A many body energy eigenstate responds to external perturbations through the flow of heat obeying the usual relationship between the change in heat and the change in entropy.
If it turns out that the entanglement entropy is not the correct measure of the thermodynamic entropy for energy eigenstates, the arguments presented in Sec.3 make it likely that there is another prescription involving solely the wave function that should determine the entropy.

Appendix A. Range of Potential Matrix Elements
In this appendix we justify in more detail the cutoff on the random matrix used in calculations. We are interested in determining how E|V |E ′ varies with increasing E − E ′ . The quantity we wish to compute is E|V |E ′ E,E ′ Here the second set of brackets denotes a microcanonical average over both Eand E ′ . V is taken to be of the form (3), and we consider a system of identical particles that are either fermions or bosons. Label the eigenstates of a single particle by i. The energy in that state is labeled e i , the total number of particles in state i is n i and n i ≤ 1 for the case of fermions. In second quantized notation the total wave-function can be written as where |0 > is the ground state. The potential V can also be written in second quantized form as and in this case, ψ j denotes a plane-wave with wave vector indexed by j. Thus the quantity we wish to compute is where n(E) is the appropriate normalization. In the above, n ′ i s means the sum over all possible combinations of n ′ i s with the constraint i n i = n. Writing the above equation in second quantized form and taking the inner products gives where the last bracket denotes a microcanonical average at energy E. As long as |E − E ′ | ≪ E, and n is large, the microcanonical average here can be replaced by a canonical average at the appropriate temperature T as given in (2.4) where If we consider potentials V (r) which have a Fourier transform that is bounded, then so is V jklm as the single particle eigenstates are plane waves. We can therefore bound the above equation by 1 n(E ′ ) j,k,l,m n ′ l ,n ′ Now consider what happens for E − E ′ ≫ T . In this limit it is straightforward to substitute in the appropriate Bose or Fermi distributions for each n i and perform the summations, but the asymptotic result can be seen by the following argument. The above average only has contributions to it when e j + e k = e l + e m + E − E ′ . If E ′ is kept fixed and E is increased then the minimum energy needed to obtain an contribution occurs when e l = e m = 0, so that e j + e k = E − E ′ .(Here we are setting the ground state energies equal to zero.) As E is increased the weight of having such a configuration is given by the appropriate Bose or Fermi distributions which asymptotically give a weight of exp(−e j − e k ) = exp(−(E − E ′ )/T ). Considering larger e l and e m does not change the above exponential dependence, but just the overall pre-factor.

Appendix B. Relation Between Entropy and Forces
Here we present a derivation of Eq. 15 that is shorter than other treatments that the author is aware of. Consider the integral of the density of states Where the last equality uses the Heaviside function θ. Because the entropy is rapidly increasing for a system with a large number of degrees of freedom n, we expand it about E, in the exponent of the above integrand, So that for large n, Now consider a Hamiltonian that depends on a parameter x, which for large n becomes ∂S/∂x, giving Eq. 15.

Appendix C. Calculation of Off Diagonal Component of Entropy
We wish to calculate the effects of small off diagonal elements of the density matrix on the entropy S(ρ) = − tr(ρ ln ρ)), by writing ρ ij = D ij + ǫ ij . To simplify the expansion, first define f (x) = −x ln(x). There is no power series expansion of this about x = 0, but we can regularize it to allow such an expansion, for example f r (x) = f (x + δ). We will see that at the end, we can take the limit δ → 0 without difficulty. With a regularized f , we can it expand it as f (x) = ∞ n=0 a n x n (C.1) S(ρ) = tr(f r (ρ)) = tr(f r (D + ǫ)) = ∞ n=0 a n (D + ǫ) n (C. 2) The ǫ 0 term yields S 0 = tr(f (D)), as expected. The ǫ 1 term is zero because tr(D n ǫ) = i D n ii ǫ ii = 0. To obtain the ǫ 2 contribution, we note that ǫ and D are in general non-commuting, and therefore we must preserve matrix ordering when expanding (D + ǫ) n . Denoting terms second order in ǫ by S 2 , we have S 2 (ρ) = tr ∞ n=0 a n ∞ i,j,k=0 D i ǫD j ǫD k δ i+j+k,n−2 (C.3) However because for two matrices A and B, tr(AB) = tr(BA), we can reorder a given term so that final matrix is always ǫ , so that the summand becomes D i+k ǫD j ǫ. This reordering generates i + k + 1 such terms. So simplifying the indices gives We now commute the trace with the summations and perform it first. It has the general form tr(AǫBǫ) with A and B diagonal and real. In this case this can easily seen to be n,m Because e is symmetric, the indices m and n can be exchanged, and so after switching the dummy variables i and j we can add this relabeled expression to the above, giving (i + j + 2)a i+j+2 x i y j (C.9) and we wish to find a closed form expression for this in terms of f r (x) which have the a n 's coefficients as it power series expansion. To do this, we group all terms according the value of n ≡ i + j. Therefore F (x, y) = ∞ n=0 (n+2)a n+2 (x n +x n−1 y+. . . xy n−1 +y n ) = ∞ n=0 (n+2)a n+2 (x n+1 − y n+1 ) x − y (C.10) This can be further simplified by noting that where the prime denotes the first derivative. Substituting this into Eq. C.10 gives Note that because of the cancellation of f ′ r (0) we can now take the limit as δ → 0 and use f = −x ln x instead. This yields F (x, y) = ln x − ln y x − y (C.13) Summarizing, we have shown that the second order correction to the entropy due to off-diagonal components of the density matrix ǫ is F (ρ nn , ρ mm )Re(ǫ 2 nm ). (C.14) [1] S. K. Ma, "Statistical Mechanics", World Scientific (1985).