A maximum entropy thermodynamics of small systemsPurushottam D. Dixit Citation: The Journal of Chemical Physics 138, 184111 (2013); doi: 10.1063/1.4804549 View online: http://dx.doi.org/10.1063/1.4804549 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/138/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Time dependent quantum thermodynamics of a coupled quantum oscillator system in a small thermalenvironment J. Chem. Phys. 139, 214108 (2013); 10.1063/1.4833566 The Second Law for Small Systems AIP Conf. Proc. 1033, 219 (2008); 10.1063/1.2979033 Statistical Wave Scattering in Chaotic and Disordered Systems: Random Matrices and Maximum Entropy AIP Conf. Proc. 757, 29 (2005); 10.1063/1.1900485 Maximum Entropy Approach to the Theory of Simple Fluids AIP Conf. Proc. 707, 17 (2004); 10.1063/1.1751353 Maximum entropy, fluctuations and priors AIP Conf. Proc. 568, 94 (2001); 10.1063/1.1381874

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THE JOURNAL OF CHEMICAL PHYSICS 138, 184111 (2013)

A maximum entropy thermodynamics of small systemsPurushottam D. Dixita)

Biosciences Department, Brookhaven National Laboratory, Upton, New York 11973, USA

(Received 12 February 2013; accepted 26 April 2013; published online 14 May 2013)

We present a maximum entropy approach to analyze the state space of a small system in contactwith a large bath, e.g., a solvated macromolecular system. For the solute, the fluctuations around themean values of observables are not negligible and the probability distribution P(r) of the state spacedepends on the intricate details of the interaction of the solute with the solvent. Here, we employa superstatistical approach: P(r) is expressed as a marginal distribution summed over the variationin β, the inverse temperature of the solute. The joint distribution P(β, r) is estimated by maximiz-ing its entropy. We also calculate the first order system-size corrections to the canonical ensembledescription of the state space. We test the development on a simple harmonic oscillator interactingwith two baths with very different chemical identities, viz., (a) Lennard-Jones particles and (b) wa-ter molecules. In both cases, our method captures the state space of the oscillator sufficiently well.Future directions and connections with traditional statistical mechanics are discussed. © 2013 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4804549]

I. INTRODUCTION

Recent developments in spectroscopic1 and singlemolecule manipulation techniques2, 3 allow direct measure-ments of very small systems. Examples of small systems fromaqueous solution chemistry and biochemistry include pro-teins/nucleic acid molecules,4 nanoparticles,5 and hydratedion [M(H2O)n]k+ complexes interacting with bulk watersolution.6 Here, we are more interested in the description ofthe dynamics within the solute system, e.g., folding of the pro-tein or interaction between nanoparticles, than the details ofits interactions with the surrounding medium. In other words,we seek an effective thermodynamic description of the sys-tem by integrating over the uninteresting “bulk” degrees offreedom.

Due to the small size of the system and the high surface-to-volume ratio, the bath of solvent particles cannot be treatedas an ideal thermal bath (one which only interacts weakly withthe system). Consequently, the system-bath interactions haveto be entertained at some detailed level of description. Weexpect the behavior of a small system to be markedly differentfrom systems with macroscopically large number of particlesdue to relatively heightened fluctuations in energy, volume,etc.7, 8

To see this clearly, consider a large system coupled to athermal bath. For simplicity, assume that the system can onlyexchange energy with the bath. Now imagine that the abovesystem can be decomposed into subsystems A and B, suchthat A is very large compared to B. For convenience, we willidentify A as the solvent and B as a solute, e.g., a protein.Throughout this article, we will use the terms “solute” and“system” and the terms “solvent” and “bath” interchangeably.The energy E of the composite system can be written as

E(rA, rB) ≡ EA(rA) + EB(rB) + EAB(rA, rB), (1)

a)Author to whom correspondence should be addressed. Electronic mail:pdixit@bnl.gov. Telephone: (631) 344-3742.

where EA (EB) is the interaction energy within the solvent(solute) and EAB is the interaction between the solvent andthe solute. rA, (rB) denote the collective coordinates of solvent(solute) particles. Note that E ≈ EA � EB(∼EAB).

In the canonical ensemble description of the system atinverse temperature β = 1/kBT, we write the distribution ofstates P(rA, rB) as

P (rA, rB|β) ∝ exp(−βE(rA, rB)). (2)

The marginal distribution for the solute degrees of free-dom is formally written as

P (rB|β)=∑rA

P (rA, rB|β) ∝ exp(−β[EB(rB) + φB(β, rB)]).

(3)

The marginal distribution depends only on rB, the inter-nal coordinates of the solute, and the inverse temperature β.Here, φB(β, rB) represents the temperature dependent effectof the solvent-solute interactions on the state space of the so-lute. Note that φB(β, rB) is a constant if the solute is verylarge, i.e., EB � EAB, and behaves independently of the sol-vent. In this case, the solute acts as a thermodynamic systemin its own right, in contact with an ideal thermal bath at thesame inverse temperature (a thermal bath comprising of idealgas particles).

Understanding the structure of the “molecular field,”φB(β, rB), is of tremendous importance, since EB(rB)+ φB(β, rB) completely describes the phase space distribu-tion of the solute.9, 10 Even though Eq. (3) is formally true,it has little practical value since φB(β, rB) depends in a non-trivial fashion on the details of the solute-solvent interactionEAB, and, in general, is quite hard to estimate.9, 11 Ad hoc as-sumptions about the structure of φB(β, rB) are commonplacein aqueous chemistry literature including (but certainly notlimited to) examples such as the generalized Born dielectricmodels12 and the nonlinear Poisson-Boltzmann model.13

0021-9606/2013/138(18)/184111/6/$30.00 © 2013 AIP Publishing LLC138, 184111-1

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184111-2 Purushottam D. Dixit J. Chem. Phys. 138, 184111 (2013)

Here, we present a maximum entropy approach to modelthe state space distribution P(rB) (see Eq. (3)) by circumvent-ing the problem of estimating the temperature dependent fieldφB(β, rB). We illustrate our framework, using molecular dy-namics simulations, to study the state space of a harmonic os-cillator (B) interacting with (a) a bath of Lennard-Jones par-ticles (A) and (b) a bath of water molecules (A), two solventmedia of completely different chemical identities.

In Sec. II, we develop our method, in Sec. III, we exploreconnections with traditional statistical mechanics, and inSec. IV, we illustrate it for a harmonic oscillator coupled totwo realistic baths. Finally in Sec. V, we discuss future di-rections and possible connections with traditional statisticalmechanics.

II. THEORY

A. Maximum entropy superstatistics

Consider a macromolecular solute interacting with a sol-vent as described above. The maximum entropy (maxEnt) in-terpretation views the problem of determining the distributionP(rA, rB) of the coordinates {rA, rB} of the composite sys-tem as an inference problem: P(rA, rB) is estimated from thelimited available knowledge of the system.14 Briefly, the prob-abilities P(rA, rB) are estimated by maximizing the entropyS[P(rA, rB)] subject to the constraint that the distribution re-produces some experimentally observed quantities such as themean energy E. The constrained objective function is givenby Eq. (4):14, 15

S[P (rA, rB)] − β

(∑rA,rB

P (rA, rB)E(rA, rB) − E

)

+ γ

(∑rA,rB

P (rA, rB) − 1

). (4)

Here, β and γ are the Lagrange multipliers and

S[P (rA, rB)] = −∑rA,rB

P (rA, rB) log P (rA, rB). (5)

The estimated distribution P(rA, rB|β) has the maximumentropy amongst all candidate distributions P∗(rA, rB) that re-produce the experimental constraints (here, average energy).P(rA, rB|β) is parameterized by a unique inverse temperatureβ and is given by Eq. (2). The maximum entropy estimate de-pends solely on the average energy E and results in a correctpredictive theory only for macroscopically large systems.14, 15

This is due to the fact that in a macroscopically large sys-tem, the fluctuation in energy, 〈E2〉–〈E〉2, is negligible com-pared to the mean 〈E〉.14 Consequently, the higher momentsof the energy distribution 〈En〉 (n > 1) can be estimated fromthe knowledge of the mean. A naive application of the max-imum entropy principle for a small system such as the so-lute is bound to result in predictions that do not match withexperiments.

Note if the solute B is sufficiently large, the internal in-teractions within the solute will vastly outweigh the solute-solvent interactions. Here, the solvent can be integrated withthe thermal bath and its effect is absent from the description

of the solute except for setting the inverse temperature β. Inthis case, the entropy of the solute S[P(rB)] itself is maxi-mized subject to constraining the average energy of the solute.Thus, for sufficiently large B, the effect of A on the phasespace of B can be represented by a single number: with re-spect to any prediction about the solute B, the equivalenceP(rA, rB) ↔ P(rB) involves minimal loss of information. Fora solute of intermediate size, we assume that the solvent canbe equivalently represented by allowing the inverse tempera-ture of the solute to fluctuate, i.e., we work with the ansatzP(rA, rB) ↔ P(β, rB) rather than P(rA, rB) ↔ P(rB). In otherwords, instead of maximizing the entropy S[P(rA, rB)] ofP(rA, rB),16 we maximize the entropy of the joint distributionP(β, rB). Thus, we maximize

S[P (β, rB)] = −∑β,rB

P (β, rB) log P (β, rB)

= S[P (β)] +∑

β

P (β)SB(β), (6)

where S[P(β)] is the entropy of

P (β) =∑rB

P (β, rB), (7)

and

SB(β) = −∑rB

P (rB|β) log P (rB|β) (8)

is the entropy of the solute in contact with an ideal thermalbath at inverse temperature β and

P (rB|β) ∝ exp(−βEB(rB)) (9)

is the canonical ensemble probability distribution. The idealthermal bath can be constructed by replacing the solvent witha dilute ideal gas solvent A′ at the same temperature such thatEB � EA′B.

In maximization, a straightforward choice for the con-straint is EB, the observed mean energy of the solute B. Sincethe entropy of the joint distribution P(β, rB) is maximizedrather than that of the marginal distribution P(rB), the mea-sured entropy SB itself becomes a constraint.17 Thus, the con-strained optimization function, including the Lagrange mul-tipliers and summing over rB degrees of freedom, is (fromEq. (6))

S[P (β)]+λ

⎛⎝∑

β

SB(β)P (β) − SB

⎞⎠+γ

⎛⎝∑

β

P (β) − 1

⎞⎠

− ζ

⎛⎝∑

β

〈EB〉βP (β) − EB

⎞⎠ . (10)

Here,

〈EB〉β =∑rB

EB(rB)P (rB|β) (11)

is the average energy of the solute when it is coupled to anideal thermal bath (A′) at an inverse temperature β.

We have transformed the problem of maximizing theentropy of the joint distribution P(β, rB) to the one of

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184111-3 Purushottam D. Dixit J. Chem. Phys. 138, 184111 (2013)

maximizing the entropy of P(β). Carrying out themaximization,

P (β) = 1

Z(λ, ζ )exp(λSB(β) − ζ 〈EB〉β), (12)

where

Z(λ, ζ ) =∑

β

exp(λSB(β) − ζ 〈EB〉β) (13)

is the generalized partition function. The marginal distributionP(rB) is written as

P (rB) =∑

β

P (rB|β)P (β). (14)

Equations (12) and (14) are the main theoretical resultsof this work. In deriving Eqs. (12) and (14), we have assumedthat the effect of the bulk solvent medium A on the solute Bcan be captured by allowing the inverse temperature of the so-lute to fluctuate. Note that the above coarse graining approachimplies that the marginal distribution P(rB) belongs to a re-stricted family of distributions. We believe that this approachwill be successful if the solute is of intermediate scale, i.e.,when 〈δφ2〉 is small but cannot be completely neglected.

B. Implicit solvation

In the development presented here, notice that the predic-tions about the solute state space (Eqs. (12) and (14)) are inde-pendent of the details of the solute-solvent interactions. Fromthe perspective of effective interaction models, the current de-velopment closely resembles an implicit solvation model.

The framework assumes that the solvent induced modula-tion in the state space of the solute is completely characterizedby allowing the temperature of the solute to vary. The distri-bution P(β) of the temperature β of the solute is governed byλ and ζ . In short, λ and ζ (along with SB(β) and 〈EB〉β ; bothare properties of the solute only) completely describe solute-solvent interactions. This is the result of the coarse grainingwhere P(rA, rB) is assumed to be equivalent to P(β, rB) withrespect to all predictions about the solute B. We believe thatEqs. (12) and (14) present an implicit solvation model thatis independent of the details of the solute-solvent chemistryand has a rigorous basis in the maximum entropy framework.We leave it for further studies to study more realistic systemsusing the current framework.

III. CONNECTIONS TO STATISTICAL MECHANICS

It is instructive to examine the limiting behavior ofEq. (12). Let us first write ζ = β0λ. The maximum of theP(β) distribution can be found by setting the first β derivativeof λSB(β) − β0λ〈EB〉β to zero. Differentiating with respect toβ, we get

−λCv(β)

β+ λβ0

Cv(β)

β2≡ 0.

It is easy to check that the second derivative is positive atβ = β0. Thus, if the heat capacity Cv(β) and λ are non-zero,the maximum of the P(β) distribution lies at β = β0.

For large λ, keeping β0 finite, P(β) will tend to a nar-rowly peaked distribution around β = β0 and can be describedby a normal distribution. The width of the normal distributionis dictated by the second derivative. If Cv(β) does not varyrapidly around β = β0, we can show that for large λ,

P (β) ∼ N(

β0,

√2β0√

λCv(β0)

). (15)

Thus, as λ → ∞, the state space of the solute B is de-scribed by a canonical ensemble distribution at inverse tem-perature β = β0 = ζ /λ. Note that as λ → ∞, β0 is also thetemperature of the surrounding bath. We conclude thatλ < ∞ captures the strength of the solute-solvent interactionsand β0 = ζ /λ is the effective temperature of the solute.

For weak solute-solvent interactions, i.e., large λ, thefirst order corrections in solute behavior due to solute-solventinteractions, i.e., the first order estimate of φ(β0, rB) (seeEq. (3)), can be computed if one knows the heat capacity ofthe solute at β0, the effective temperature of the solute. Drop-ping the β0 dependence of Cv for brevity and writing P(rB):

P (rB) =∫

P (β)P (rB|β)dβ

≈∫ ∞

0N

(β0,

√2β0√

λCv(β0)

)

× exp(−βE(rB) + βF (β0))dβ, (16)

∝ exp(−β0E(rB))exp

(β2

0E(rB)2

λCv

)

×(

1 + Erf

[λCv − 2β0E(rB)

2√

Cvλ

]). (17)

From Eq. (17), by expanding for energies small com-pared to λCv , we estimate φ(β0, rB) as (see Eq. (3) for thedefinition of φ)

φ(β0, rB) ≈ e− Cvλ4 E(rB)√πCvλ

− β0E(rB)2

Cvλ. (18)

As expected, φ(β0, rB) ∼ 0 when λ → ∞. It will beinteresting to explore if Eqs. (17) and (18) can be exploitedin molecular dynamics simulations. We leave it for furtherstudies.

Even under the simplifying condition of weak solute-solvent interactions, i.e., large λ, the maximum entropyapproach suggests that the solvent modulates the phasespace behavior of the solute in a highly non-trivial manner.Equations (17) and (18) provide a glimpse into the mecha-nism of the solvent-induced modulation of the phase space ofthe solute. The first term effectively decreases its temperaturewhile the second term increases the propensity to sample highenergy states thus effectively increasing the temperature.

Notice that the phase space described by Eq. (18) is iden-tical to a maxEnt probability distribution where the mean en-ergy 〈E〉 and the fluctuation in the energy 〈E2〉 are constrained(see Eqs. (18) and (3)). Previously, the predictions fromEq. (18) have been validated for ferromagnetic materials.18, 19

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184111-4 Purushottam D. Dixit J. Chem. Phys. 138, 184111 (2013)

In the current framework, however, instead of constrainingthe higher moments of energy, we have constrained the av-erage entropy of the small system. The average entropy isa natural constraint for small systems: due to non-negligiblesystem-bath interactions, unlike for a thermodynamic system,the entropy of the distribution P(rB) of the phase space ofthe small system is the one with the maximum entropy underthe constraint of mean energy. We speculate that if the ansatzP(rA, rB) ↔ P(β, rB) (see above) is violated, constraining thehigher moments of the energy may prove useful.

IV. HARMONIC OSCILLATOR

A. Theory

We analytically illustrate the above development(Eqs. (12), (14), and (18)), using molecular dynamicssimulations,20 for a harmonic oscillator, comprising oftwo Lennard-Jones particles, coupled to a solvent bathof (a) hydrophobic Lennard-Jones particles and (b) watermolecules.

Imagine a harmonic oscillator solute (B) interacting witha bath of solvent particles (A) at inverse temperature β. With-out loss of generality, let the energy of the oscillator beE(r) = r2. If the interactions of the oscillator with the sur-rounding particles are very weak, we know that the probabil-ity distribution P(r) is parametrized by the inverse tempera-ture and is given by

P (r|β) = 4r2β3/2e−r2β

√π

. (19)

Note that for the harmonic oscillator in contact with aweakly interacting bath, the average energy 〈E〉β ∼ 1/β andthe entropy S(β) ∼ log β up to an additive and multiplicativeconstant. If the oscillator-solvent interactions are not weak,Eq. (19) does not adequately describe P(r) (see Fig. 2). In thiscase, as discussed above, we allow the inverse temperature β

of the oscillator to fluctuate. From Eq. (12),

P (β) ∝ exp

(λ log β − ζ

β

)

= e− ζ

β β−λζ λ−1

(λ − 1). (20)

Before we derive the marginal distribution P(r), let us in-spect the limiting behavior of P(β). Figure 1 shows P(β) fordifferent values of λ and ζ , keeping β0 = ζ /λ fixed. Observethat at fixed β0, P(β) distribution with higher values of λ ispeaked and becomes a dirac delta distribution δ(β − ζ /λ).Thus, as seen above, while β0 = ζ /λ represents the temper-ature of the solute, λ characterizes the strength of its inter-actions with the solvent: higher λ implies weaker interac-tions. We hypothesize that in an experiment where the solventcomposition is fixed, λ cannot be changed smoothly, whileβ0 = ζ /λ can be tuned by changing the temperature ofthe bath; however, in case of an aqueous solvent, λ can bechanged by perturbing the solute-solvent interactions usingosmolytes and salts.

FIG. 1. The distribution P(β) of the temperature of the harmonic oscillatorsolute. As λ → ∞, P(β), the distribution of β, approaches a Dirac delta dis-tribution. Notice that at small values of λ, the P(β) distribution is very broad.Hence, while ζ /λ is equivalent to the temperature of the solute, λ captures thestrength of the interactions between the solvent and the solute.

Finally, the marginal distribution P(r) = ∫P(β) ·

P(r|β)dβ is given by

P (r) =8rλ− 1

2 ζλ2 + 1

4 Kλ− 52(2r

√ζ )

√π(λ − 1)

. (21)

Here, Kγ (x) is the modified Bessel function of the second kindwith parameter γ . To understand Eq. (21) physically, let uswrite ζ = β0λ and calculate the average energy,

〈E〉 = 〈r2〉 = 3(λ − 1)

2β0λ. (22)

As λ → ∞,

〈E〉 = 3

2β0. (23)

Hence, as λ → ∞, the oscillator behaves as if it is in con-tact with an ideal thermal bath at inverse temperature 2β0/3.Similarly, it is easy to check that the limiting behavior ofEq. (21) is equal to that of Eq. (12), the distribution of statesof a harmonic oscillator in contact with an ideal thermal bath.

B. Numerical simulation

An interesting aspect of the development above is thatit hypothesizes that the form of the probability distributionP(rB) is independent of the chemistry of the solute-solventinteractions. The solvent medium affects the solute onlythrough the “coupling parameter” λ (see Eqs. (12) and (14)).We test this prediction by studying a harmonic oscillator in-teracting with two systems that are chemically very differentfrom each other, viz., a bath of Lennard-Jones particles and abath of water molecules (see the Appendix for details).

Figure 2 shows the observed probability distribution(black circles) from the molecular dynamics simulations,the best-fit canonical ensemble distribution (black line, seeEq. (19)), and the best-fit of Eq. (21) (see the Appendix forthe fitting procedure). Note that even though these two solventsystems have very different chemical identities, Fig. 2 clearlyshows that the empirically observed distribution for the twosystems has an extended tail that cannot be captured by thecanonical ensemble distribution. Meanwhile, the distribution

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184111-5 Purushottam D. Dixit J. Chem. Phys. 138, 184111 (2013)

FIG. 2. The experimentally observed P(r) distribution (black circles) compared with the best-fit canonical ensemble distribution (black line, Eq. (19))and the best-fit of Eq. (21) (red line). Left panel shows the harmonic oscillator interacting with water molecules and the right panel shows the oscilla-tor interacting with Lennard-Jones particles. Note that these two solvent systems have very different chemical identities. Yet, Eq. (21) describes the ex-perimentally observed distribution over 5 orders of magnitude especially in the extended tail that the canonical ensemble distribution Eq. (19) fails tocapture.

P(r) of Eq. (21) captures the empirically observed distributionextremely well.

V. DISCUSSION

In traditional statistical mechanics, the inverse tempera-ture β is an intensive variable and its fluctuation does not havea direct physical interpretation. Here, the entropy is relatedto heat loss and is a measurable quantity. On the other hand,the maximum entropy interpretation of statistical mechanicsfinds its basis in the information theoretic interpretation ofentropy,14, 15, 21 where the inverse temperature β is a Lagrangemultiplier and entropy is an inference tool. Within the maxEntframework β can be treated as a parameter for the canonicalensemble distribution P(rB|β). Fluctuations in β can be in-terpreted as arising due to the hyper-probability P(β) in theparameter space.22

The maxEnt interpretation views statistical mechanics asan inference problem: among candidate distributions P∗(rB)that reproduce the known experimental measurements aboutthe system B, maxEnt states that the one which has the max-imum entropy describes the internal states of the system cor-rectly. For a system exchanging energy with the bath, thesuccess of maxEnt is heavily dependent on the assumptionof weak system-bath interactions (i.e., weak solute-solventinteractions).14, 15 Here, we have shown that the maxEnt inter-pretation also offers a way to generalize statistical mechanics.

Note that regardless of the size of the system B, the en-tropy of the joint system comprising of B and the bath A ismaximized.16 When the interactions between A and B are nottoo strong (but not too weak to be negligible either), we hy-pothesized that the effect of the variation in the states rA ofthe solvent bath on the solute can be characterized by vary-ing its inverse temperature β. In other words, we assumedthat the equivalence P(rA, rB) ↔ P(β, rB) involved minimalloss of information about the bath. Our development leadto a generalization of statistical mechanics that resemblessuperstatistics.17, 23 We established a direct correspondencebetween the developed framework and traditional statisticalmechanics in the limiting case of very weak solute-solventinteractions. We also calculated the first order correction to

canonical ensemble description of the state space of smallsystems.

We illustrated the framework with a harmonic oscillatorcoupled to two solvent baths of very different chemical iden-tities. We showed that the superstatistical distribution Eq. (21)describes the distribution of states P(r) of the oscillator betterthan the usual canonical ensemble distribution Eq. (19). Thecurrent development has shown that the thermodynamics ofsmall systems can be suitably described by a superstatistics.We believe that the framework will be useful in understandingsolvent induced modulations of solute state space, e.g., withimplicit solvent models.

ACKNOWLEDGMENTS

I would like to thank Professor Ken Dill, Professor DilipAsthagiri, Professor German Drazer, Professor Steve Pressé,and Mr. Sumedh Risbud for stimulating conversations andsuggestions about the article.

This work was supported by grants (Grant No. PM-031)from the Office of Biological Research of the U.S. Depart-ment of Energy.

APPENDIX A: NUMERICAL SIMULATION

A harmonic spring consisting of two Lennard-Jones par-ticles was immersed in a bath of 512 Lennard-Jones parti-cles in a cube of side 25 Å and a bath of 256 TIP324, 25

water molecules at 300 K. NVT molecular dynamics simu-lations were run with NAMD20 at 300 K. The CHARMM26

forcefield was used to describe the interaction between theLennard-Jones particles and between the spring and the bathof particles. The spring constant for the harmonic oscillatorwas chosen to be k = 0.5 kcal/mol Å2. The ε parameter forthe Lennard-Jones bath was set at −0.015, while the ε param-eter for the spring was set at ε = −10.0. The size parameterwas set at r = 2.1 Å for the oscillator particles (r = 1.1 Åwhen interacting with water) and r = 1.1 Å for the bath parti-cles. The systems were minimized for 2000 steps followed byan equilibration of 1 ns and a production run of 2 ns. Config-urations were stored every 100 fs.

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184111-6 Purushottam D. Dixit J. Chem. Phys. 138, 184111 (2013)

APPENDIX B: FITTING THE PARAMETERS

In order to fit Eq. (21) to the experimental data, one needsto determine the free parameters from the simulation. In thetraditional canonical ensemble, the inverse temperature β ofthe harmonic oscillator will be estimated from its average en-ergy. Here, we show how to estimate the free parameters fromthe simulation. It is non-trivial to measure the average systementropy 〈SB(ζ )〉 in a computer simulation. Yet, operationally,

〈SB(β)〉 ∝∫

log β · P (β)dβ

∝∫ ∫

log r · P (r|β)P (β)drdβ

=∫

〈log r〉βP (β)dβ. (B1)

In other words, constraining SB(β) is equivalent to constrain-ing log r. Thus, in addition to 〈E〉 = 〈r2〉, we also estimate〈log r〉 from the simulation and then fit Eq. (21).

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