q-exponential distribution in time correlation function of water ...

Brazilian Journal of Physics, vol. 39, no. 2A, August, 2009

439

q-exponential distribution in time correlation function of water hydrogen bonds M. G. Campo, G. L. Ferri, G. B. Roston

Departamento de F´ısica, Universidad Nacional de la Pampa Uruguay 151,(6300) Santa Rosa, La Pampa, Argentina (Received on 23 December, 2008) In a series of molecular dynamics simulations we analyzed structural and dynamics properties of water at different temperatures (213 K to 360 K), using the Simple Point Charge-Extended (SPC/E) water. We detected a q-exponential behavior in the history-dependent bond correlation function of hydrogen bonds. We found that q increases with T −1 below approximately 300 K and is correlated to the increase of the tetrahedral structure of water and the subdiffusive motion of the molecules. Keywords: Water hydrogen bonds, q-statistics, Cage effect

1.

INTRODUCTION

Water physical properties have been investigated in far greater details, because water is one of the most ubiquitous substances on Earth and most life processes are related with their properties [1–3]. In particular, structural and dynamic properties of the hydrogen bonds have been investigated using different experimental and theoretical techniques, nevertheless several issues remain unsolved. X-Ray diffraction, neutron diffraction and proton magnetic shielding tensor measurements can be related to the oxygen-oxygen radial distribution, hydrogen bonds geometry of water and other structural properties [4–6]. A number of experimental and simulation techniques were used to study the elemental dynamics process of rupturing and forming of hydrogen bonds in water, characterized by an average bond lifetime τHB and caused by diffusion and librational motions of water molecules on a very fast time scale [7–13]. In molecular dynamics (MD), a “history-dependent” bond correlation function P(t) is used to obtain τHB [14, 15]. P(t) represents the probability that an hydrogen bond formed at a time t = 0 remained continuously unbroken and will be broken at time t.

τHB =

Z∞

tP(t)dt

(1)

0

Both MD simulations and experimental data for depolarizated light scattering show an Arrhenious behavior of τHB , where τHB ∝ exp(−E ∗ /RT ) and E ∗ is an activation energy [13–15]. E ∗ can be interpreted as the energy required to break a hydrogen bond (≈ 10 KJ/mol). However, the functional behavior of P(t) has not been clearly established to date. In previous works it was found that P(t) not to verify either an exponential or a power law behavior in the liquid and supercooled region of water, regardless of the hydrogen bond definition used [14, 15]. In other works the dynamics of liquid and supercooled water was studied by analyzing the behavior of the mean-square displacement time series (M(t) ) M(t) = h[r(t) − r(0)]2 i

(2)

where r(t) is the position of de oxygen atoms of water at time t, r(0) is their initial position and the brackets hi denote ensemble average. At room temperature the motion of

water molecules at short times is ballistic, where M(t) ∝ t α and α ≈ 2 for pure ballistic motion. At later times the water molecules move diffusively (α ≈ 1 according to Einstein relation) and the diffusion coefficient D can be calculated. However, a plateau emerges in M(t) (0 < α < 1) at intermediate times in the supercooled region of water (T . 273 K) where the movement of the water molecules is called subdiffusive [16, 17]. This phenomenon is attributed to a caging behavior of water, in which a water molecule is temporarily trapped by its neighbors and then moves in short burst due to nearby cooperative motion. The diffusion is associated with a non-Gaussian statistics, when the Einstein relation is not satisfied [18–20]. In particular, the caging behavior of water is studied by Mazza et. al [17] calculating the non-Gaussian parameter α2 (∆t) of the displacement of water molecules. Constantino Tsallis and collaborators introduced the qexponential probability distribution [21]. This can be defined through their ”complementary” distribution functions, also called ”survival ” functions:   (1 − q) x 1/(1−q) Pq,κ (X > x) = 1 − κ

(3)

Tsallis et al. proposed these distributions to handle statistical-mechanical systems with long-range interactions, necessitating a non-extensive generalization of the ordinary Gibbs-Shannon entropy. Following Jaynes’s procedure [22] of maximizing an entropy subject to constraints on expectation values, they got the q-exponential ditributions, in wich κ enforces the constrains, and q measures the departure from extensivity, Boltzmann-Gibbs statistics being recovered as q → 1. Sometimes, a q-exponential behavior of dynamic variables has been observed in cases of particles with a subdifussive behavior [18, 23, 24]. However, this last behavior has not been considered in water dynamic variables of molecular dynamics simulations. In this work, using a MD method, we studied the structural and dynamics behavior of the extended simple point charge (SPC/E) model of water over a wide range of temperatures (213-360 K). As a result, we will show that the bond correlation function, P(t), has a non-exponential behavior and fits to a q-exponential function. Besides, we analyzed possible correlations between this q-exponential behavior of P(t), the hydrogen bonds distributions and the displacement of water molecules.

were 360, 343, 323, 313, 303, 293, 283, 273, 263, 253, 233 and 213 K. We assigned the velocities of the molecules 440 M. G. according Campo ettoal. Boltzmann’s distribution in the system at 360 K. The equili273 1.11 0.02 0.994 -4.125 -1.243 353.3 bration method was similar for all systems: initially, we made 283 1.10 0.02 0.994 -4.309 -1.211 325.2 a 200 steps of ∼ energy continuing with energy a equi- as libration run of 1 ns minimization, using the constant potential TABLE I: Fitting293 parameters of 0.993 the bond-4.992 correlation function 1.11 0.02 -1.047 286.4correlibrationcriterion. run of ∼ 1Then, ns using constant the potential energyfor as an stability we the calculated trajectory sponding to the different simulated systems. stability criterion. Then, we calculated the trajectory for an 303 1.10 0.02 0.993 -5.174 -1.025 266.3 additional 3 ns. The preceding higher temperature liquid conT / K q ∆q r2 a . 103 fs−1 b τHB / fs additionalwas 3 ns.used The as preceding higherpoint temperature liquid configuration the starting for each successive 313 1.09 0.02 0.992 -5.412 -0.985 245.3 213 1.28 0.02 0.997 -4.798 -0.780 576.4 figuration was as of thethe starting pointwere for each successive simulation. The used values densities according to the 323 1.09 0.02 0.991 -5.555 -0.975 238.8 233 1.23 0.01 0.997 -4.840 -0.842 456.7 simulation. The values of the densities were according the 2 bibliography [16, 29, 30]. The simulation time stepto was 343 1.08 0.02 0.993 -6.300 -0.855 201.8 253 1.16 0.02 0.996 -4.457 -1.068 386.3 bibliography [16, 29, 30]. The simulation time step was 2 fs and the trajectories were collected every 10 fs for all sim3600.01 1.060.996 0.01 0.995 -0.884 188.7 263 1.13 -4.280-6.342 -1.149 362.8 fs and the trajectories were collected every 10 fs for all simulations. The sampling time is shorter than the typical time 273 1.11 0.02 0.994 -4.125 -1.243 353.3 ulations. The sampling time is shorter than the typical time during which canbe bedestroyed destroyedbyby libration 283 1.10 0.02 0.994 -4.309 -1.211 325.2 during whichaahydrogen hydrogen bond bond can libration movements. to a q-exponential function. Besides, we analyzed 293 1.11 0.02 0.993 -4.992 -1.047 286.4 possible movements. correlations between q-exponential behavior 303 1.10 0.02 0.993this -5.174 -1.025 266.3of P(t), the hydrogen distributions and the displacement 313 1.09 bonds 0.02 0.992 -5.412 -0.985 245.3 of water q molecules. 323 1.09 0.02 0.991 -5.555 -0.975 238.8 1.3 343 1.08 0.02 0.993 -6.300 -0.855 201.8 360 1.06 0.01 0.995 -6.342 -0.884 188.7 233 1.23 0.01 0.997

-4.840

-0.842 456.7

253 1.16 0.02 0.996 263 1.13 0.01 0.996

-4.457 -4.280

-1.068 386.3 -1.149 362.8

2. THEORY AND METHOD

2. THEORY AND METHOD We performed MD simulations of the SPC/E model of water using the GROMACS package [25, 26]. This water model assumed a MD rigidsimulations geometry of of thethe molecules, with anof O–H We performed SPC/E model wa-dis◦ . The hydrogen tance of 0.1 nm and a H–O–H angle of 109.47 ter using the GROMACS package [25, 26]. This water model charge is qH =of0.4238e and the oxygen charge assumedatom a rigid geometry the molecules, with anatom O–H dis- is −2q . The parameters of the Lennard-Jones interactions ◦ tance of 0.1Hnm and a H–O–H angle of 109.47 . The hydro-between the oxygen atoms were those of the AMBER-99φ force gen atom charge is qH = 0.4238e and the oxygen atom charge field [27]. logq(P(t))

is −2qH . The parameters of the Lennard-Jones interactions between the oxygen atoms were those of the AMBER-99φ force field [27]. 0 -2

323 K

1.2

1.1

1.0 2.5

3.0

3.5

4.0 -3

4.5

5.0

-1

1/T /10 K

283 K

FIG. 2: Non-extensivity parameters q values as a function of 1/T .

The2:possible significanceparameters of the change of q behavior with T isofdisFIG. Non-extensivity q values as a function 1/T . cussed in thesignificance text. The possible of the change of q behavior with T is discussed in the text. -6 We used a geometric definition of hydrogen bond [10] with aWe maximum distance between their of oxygen atomsbond of 3.5[10] A˚ and used a geometric definition hydrogen with -8 145◦ as thedistance minimum angle formed by the atoms donor a maximum between their oxygen atomsOof 3.5–H– A˚ and O◦acceptor . In order to obtain the hydrogen bonds distribution 145 as the minimum angle formed by the atoms Odonor –H– -10 function, each to simulation, wehydrogen calculatedbonds a histogram of Oacceptor . Ininorder obtain the distribution the quantity of hydrogen bonds for each analyzed molecule function, in each simulation, we calculated a histogram of -12 then normalized it withbonds the number of trajectories the theand quantity of hydrogen for each analyzed and molecule 0.0 0.5 1.0 1.5 2.0 2.5 number of water molecules in the system. Then, we obtained and then normalized it with the number of trajectories and the t / ps f (n) with n = 1, 2, ..., 5, being f (n) the probability of occurnumber of water molecules in the system. Then, we obtained rence of n hydrogen bonds. f (n)The withbond n =correlation 0, 1, ..., 5,function being fP(t) (n) the of simoccurwasprobability obtained from of n hydrogen bonds. FIG. 1: Fitting of P(t) of the systems at to 283 (¤) and 323 K (O). rence ulations by building a histogram of the hydrogen bonds lifeThe blue lines correspond to the calculated P(t). The bond correlation function P(t)wewas obtained fromfolsimtimes for each configuration. Then, analyzed if P(t) -4

FIG. 1: Fitting of P(t) of the systems to 283 () and 323 K (O). The lines correspond to the calculated P(t).

We carried out MD simulations with a constant number of water molecules (1158), in which we used the Berendsen’s thermostat to apply a thermal and a hydrostatic bath to the system, obtaining isobaric-isothermal ensembles at 1 atm of pressure [28]. The values of the systems’ temperatures were 360, 343, 323, 313, 303, 293, 283, 273, 263, 253, 233 and 213 K. We assigned the velocities of the molecules according to Boltzmann’s distribution in the system at 360 K. The equilibration method was similar for all systems: initially, we made a 200 steps of energy minimization, continuing with an equi-

ulations by building a histogram of the hydrogen bonds’ lifetimes for each configuration. Then, we analyzed if P(t) follows a Tsallis distribution of the form expq (t) = [1 + (1 − q)t]1/(1−q)

(4)

being t the hydrogen bonds lifetime and q the nonextensivity parameter. In the limit when q → 1, expq (t) → exp(t). The non-Gaussian behavior of the displacement of water molecules was studied calculating the time t ∗ , the time at which the non-Gaussian parameter α2 (t) reaches a maximum. The non-Gaussian parameter is

 3 r4 (t) α2 (t) = −1 5 hr2 (t)i

(5)

(4)

being t the hydrogen bonds lifetime the August, nonextensivity Brazilian Journal of Physics, vol. 39,and no.q2A, 2009

ps

parameter. In the limit when q → 1, expq (t) → exp(t).

270 K

Occurrence percentag

expq (t) = [1 − (1 − q)t]

4 HB

40

5 HB

441

30

20

320 K

10

0

0

210

10

240

270

300

330

360

T / K

-1

10

0.8

1.0 T

/T

MD

/K

1.2

-1

Occurrence percentage of HB

0

(b)

4 HB

0

2 HB 1 HB

0

300 K

0

0

0

3.0

3.5

1/ T / 10

-3

4.0 -1

4.5

K

FIG. 3: Average bond lifetime τHB from Montrose et al. depolar-

FIG. 3: light Average bondexperiments lifetime τHB from et al. depolarizated scattering [13] (°)Montrose and from our simulation izated light scattering experiments [13] ( ) and from our simulation (¤). To facilitate comparison of the results, we scaled the temper(). To facilitate comparisonSPC/E of the results, we scaled the temperaature of our simulation by TSPC/E = 245 K, and the temperature of MD HMD ture of our simulation by T = 245 and the temperature of 2O the experimental data by TMD = 277 K (TK, MD : Temperature at which H2 O the by TMD = 277 K (TMD : Temperature at which theexperimental water densitydata displays a maximum). the water density displays a maximum). The non-Gaussian behavior of the displacement of water molecules

4 was  studied

calculating the time t ∗ , the time at where r (t) and r2 (t) parameter are the fourth second moments which the non-Gaussian α2 (t) and reaches a maximum. ofThe thenon-Gaussian displacementparameter distribution, is respectively. α2 (t) is known

q 1.3

(c)

1.2

273 K

1.1

to be zero for a Gaussian distribution [17, 31].

­ ® 3 r4 (t) α2 (t) = −1 (5) 3. RESULTS AND DISCUSSION 5 hr2 (t)i ­ 4 ® ­ 2 ® where r (t)theand r (t) q-exponential are the fourth behavior and second To study possible of moments the bond of the displacement distribution, respectively. α (t) known 2 correlation function we followed this procedure: Weisassigned to be zero for a Gaussian distribution [17, 31]. successive values to q ( 1 < q . 1.3, with a step of 0.01) and

made a linear fitting 1 of logq P(t), choosing the q value with the highest correlation coefficient r2 . Below the 130 fs the be3. RESULTS AND DISCUSSION havior of P(t) is strongly influenced by the libration of atoms. Above 2.5 ps the statistics is poor, because of the unlikely To study the possible q-exponential behavior of the bond fact that two water molecules remain bonded by a hydrogen correlation function we followed this procedure: We assigned bond at this time. Hence, we fitted all the P(t) functions in successive values to q ( 1 < q . 1.3, with a step of 0.01) and the interval from 1301 fs fs, finding that P(t) can be made a linear fitting of to log2500 q P(t), choosing the q value with acceptably fitted with a q-exponential and may be written as follows 1

logq (x) =

1−x1−q

1

P(t) = P01−q expq (at + b) = P0 [1 + (1 − q)(at + b)] 1−q

(6)

where P0 , a and b are constants. The values of q and others parameters of the fitting for different temperatures are indicated in Table I. Fig. 1 shows, as an example, the P(t) fitting of the systems at 283 and 323 K. Fig. 2 shows that q ∝ T −1 for temperatures below ∼300 K, whereas this parameter is ∼1 for temperatures above this value. This fact indicates that a clear non-exponential behavior of P(t) occurs in the region below approximately 300 K.

1

logq (x) =

1−x1−q 1−q

1.0 20

30

40

50

Occurrence percentage of HB

FIG. 4: (a) Hydrogen bonds distribution functions for n=1 to n=5 versus T . (b) Hydrogen bonds distribution functions for n=1,2 and 4 versus T −1 . (c) q versus probability of occurrence of four hydrogen bonds.

the highest correlation coefficient r2 . Below the 130 fs the behavior of P(t) is strongly influenced by the libration of atoms. Above the 2.5 ps the statistical is poor, because of the unlikely FIG.that 4: two (a) Hydrogen bonds distribution functions n=0 to n=5 fact water molecules remain bonded by for a hydrogen versusatTthis . (b)time. Hydrogen bonds distribution functions for n=1,2 and bond Hence, we fitted all the P(t) functions in 4 versus T −1 . from (c) q versus occurrence fourcan hydrogen the interval 130 fsprobability to 2500 fs,of finding thatofP(t) be bonds. acceptably fitted with a q-exponential and may be written as follows

The q values higher than one indicate the increase of the probability that two molecules remain bonded much longer. Although the average bond lifetimes τHB obtained using Eq. 1 and 6 do not coincide with those of experimental data from polarizated light scattering experiments (see Fig. 3), both curves can be fitted by an Arrhenius behavior (τHB ∝ exp(−E ∗ /RT )). This fact was observed in previous works [14] [15]. The hydrogen bonds distribution functions for different temperatures are shown in Fig. 4(a). In this graphic we can distinguish three regions : The first one between 213 and ∼270 K in which predominates the four hydrogen bonds

1.4

q

(b)

M. G. Campo et al.

1.3

M(t) / nm

2

from polarizated light scattering experiments (see Fig. 3), both curves can be fitted by an Arrhenius behavior (τHB ∝ 442 exp(−E ∗ /RT )). This fact was observed in previous works [14] [15].

1.2 0

10

1.1

-1

10

1 1

10

t

-2

*

/ ps

f(4 HB)

10

(c) 1.2

-3

10

1

t / ps

10

100 0.8

FIG. 5: Mean square displacement of water in the systems at 213 K (solid line), 273 K (dashed line) and 360 K (dotted line).

FIG. 5: Mean square displacement of water in the systems at 213 K (solid line), 273 K (dashed line) and 360 K (dotted line). 0.4 The hydrogen bonds distribution functions for different temperatures are shown in Fig. 4(a). In this graphic we can distinguish three regions : A first one between 213 and ∼270 20 30 40 50 K in which (predominates four hydrogen bonds probability f (4)) that isthehigher than f (3) andprobability f (2), a sect* / ps ( f (4)) that is higher than f (3)and and∼320 f (2), aK, second region beond region between ∼270 in which f (3) > tween andfinally, ∼320 K, in which f (3) > f (4) > f (2) f (4) > f∼270 (2) and a third region above ∼320 K inand which finally, third in which f (3)regime > f (2)almost > FIG. 6: (a) time when α2 (t) reaches a maximum t ∗ vs. 1/T . (b) f (3) > fa(2) > region f (4). above In the∼320 lowerKtemperature 1 f (4). In the lower temperature regime almost 50 % of wa50 % of water molecules have four hydrogen bonds, this per-Non-extensivity parameter q vs. t ∗ . (c) g(t ∗ ) vs f (4), g(t ∗ ) = (t ∗ )− 4 ter molecules have four hydrogen bonds, this percentage decentage decreases when the temperature increases, reaching. creases when the temperature increases, reaching 20 % when 20 % when T = 360 K. In Fig. 4(b) we can observe the ocT = 360 K. In Fig. 4(b) we can observe the occurrence of a currence of relation a reciprocal relation (2),and f (1) reciprocal between f (2), between f (1) and ff(4) the and tem- f (4) To study the correlation between the structural transition and the temperature. These results coincide only qualitatively and the change of q we analyzed the behavior of q versus f (4) perature. These results coincide only qualitatively with the with ones by obtained Sutmann and According Vallauri [32]. onesthe obtained Sutmannbyand Vallauri [32]. to the Ac-(see Fig. 4 (c)). We can observe that for temperatures below cording to theand water model and thedefinition hydrogenused, bond definition300 K there are a linear correlation between both variables. water model the hydrogen bond amongst used, other things, results from different simula-In other words, the produced structural change below approxotheramongst things, the results from the different simulations may quantions may differ. quantitatively differ. imately 320 K causes the change in the statistical behavior titatively

To study the correlation between the structural transition and the change of q we analyzed the behavior of q versus f (4) (see Fig. 4 (c)). We can observe that for temperatures below 300 K there is a linear correlation between both variables. In other words, the produced structural change below approximately 320 K causes the change in the statistical behavior of P(t) and, as a consequence, a significative increase of the probability of two water molecules be bonded much longer. Fig. 5 shows the mean square displacements for the systems at 213, 273 and 360 K. M(t) shows some evidence of the caging behavior for water at temperatures below 300 K, where we can observe the appearance of an intermediate plateau at times between 0.1 and 100 ps that indicates a subdiffusive motion of the water. At longer times, there is a recovery of diffusive motion. We can observe from Fig. 6 (a) that t ∗ increases with the decrease of T , according to the results obtained by Mazza et al. [17]. Clearly, t ∗ ∼ T −1 at temperatures below 300 K. This fact is attributable to the increase of the subdiffusive motion of the water. Fig. 6 (b) shows that q and t ∗ are correlated when the temperature drops below 300 K, which is an expected result because both parameters have a similar behavior with this variable. These changes in q and t ∗ are produced by a modification in the water structure when f (4) outnumbers f (2) at

FIG. 6: (a) time when α2 (t) reaches a maximum t ∗ vs. 1/T . (b) 1 Non-extensivity parameter q vs. t ∗ . (c) f (4)− 4 vs t ∗ .

approximately 320 K, and a structure in which prevails three and four hydrogen bonds begins to be the most frequent one in the water. Moreover, in Fig. 6 (c) we can see the correlation between t ∗ and f (4) for values corresponding to the systems below 1 300 K, in which it is observed that t ∗ ∼ f (4)− 4 . Considering these results, the four hydrogen bonds probability associated to a tetrahedral structure in water is a critical variable in the phenomenon of ”cage effect” that occurs at

Brazilian Journal of Physics, vol. 39, no. 2A, August, 2009 low temperatures and the dynamics of the hydrogen bonds that changes from an exponential to a non-exponential distribution. 4.

CONCLUSIONS

We have shown that the temporal correlation function of hydrogen bonds P(t) has a q-exponential behavior. The nonextensivity parameter q takes values above 1 below approximately 300 K. This increase of q indicates the increase of the probability that two molecules remain bonded during a long time t. The transition of P(t) from an exponential to

[1] D. Eisenberg and W. Kauzmann, The Structure and Properties of Water (Clarendon Press, Oxford, 1969). [2] F. H. Stillinger, Science 209, 451 (1980). [3] O. Mishima and H. E. Stanley, Nature (London) 396, 329 (1998). [4] A. H. Narten, M. D. Danford and H. A. Levy, Faraday Discuss., 43, 97 (1967). [5] A.K. Soper, F. Bruni and M. A. Ricci, J. Chem. Phys. 106, 247 (1997). [6] K. Modig, B. G. Pfrommer and B. Halle, Phys. Rev. Lett., 90(7), 075502 (2003). [7] C.A. Angell and V. Rodgers, J. Chem. Phys. 80, 6245 (1984). [8] J.D. Cruzan, L.B. Braly, K. Liu, M.G. Brown, J.G. Loeser and R.J. Saykally, Science 271, 59 (1996). [9] S. Woutersen, U. Emmerichs and H. Bakker, Science 278, 658 (1997). [10] A. Luzar and D. Chandler, Phys. Rev. Lett. 76, 928 (1996). [11] A. Luzar, J. Chem. Phys. 113, 10663 (2000). [12] F. Mallamace, M. Broccio, C. Corsaro, A. Faraone, U. Wandrlingh, L. Liu, C. Mou, and S.H. Chen, J. Chem. Phys. 124, 124 (2006). [13] C.J. Montrose, J.A. B´ucaro, J. Marshall-Coakley and T.A. Litovitz, J. Chem. Phys. 60, 5025 (1974). [14] F.W. Starr, J.K. Nielsen, and H.E. Stanley, Phys. Rev. Lett. 82, 2294 (1999). [15] F.W. Starr, J.K. Nielsen, and H.E. Stanley, Phys. Rev. E. 62, 579 (2000).

443 a q-exponential behavior occurs in parallel with a structural modification in water, where the probability of occurrence of four hydrogen bonds outnumbers the one of two hydrogen bonds. The increase of q is also correlated with the increase of the Non-Gaussian behavior of water displacement. This fact is associated which the increase of t ∗ , the time at which the Non-Gaussian parameter α2 (∆t) reaches a maximum. Acknowledgment M.G. Campo, G.L. Ferri and G.B.Roston are grateful for the financial support by PICTO UNLPAM 2005 30807 and the Facultad de Ciencias Exactas de la UNLPam.

[16] S. Chatterjee, P.G. Debenedetti, F.H. Stillinger and R.M. Lynden-bell, J. Chem. Phys. 128, 124511 (2008). [17] M.G. Mazza, N. Giovambattista, H.E. Stanley and F.W. Starr, Phys. Rev. E 76, 031203 (2007). [18] B. Liu and J. Goree, Phys. Rev. Lett. 100, 055003 (2008). [19] T. H. Solomon, E. R. Weeks, and H. L. Swinney, Phys. Rev. Lett. 71, 3975 (1993). [20] S. Ratynskaia et al., Phys. Rev. Lett. 96, 105010 (2006). [21] C. Tsallis, J. of Statistical Phys. 52, 479 (1988). [22] E. T. Jaynes, Essays on Probability, Statistics, and Statistical Physics (Reidel, London, 1983). [23] C. Tsallis et al., Phys. Rev. Lett. 75, 3589 (1995). [24] C. Tsallis, Brazilian J. of Phys., 29(1), (1999). [25] H.J.C.Berendsen, J.R. Grigera, T.P. Straatsma, J. Phys.Chem. 91, 6269 (1987). [26] H.J.C. Berendsen, D. van der Spoel and R.V. Drunen, Comp. Phys. Comm. 91, 43 (1995). [27] E.J. Sorin and V.S. Pande, Biophys. J. 88 (4), 2472 (2005). [28] H.J.C. Berendsen, J. Postma, W. van Gusteren, A. Di Nola and J. Haak, J. Chem. Phys. 81, 3684 (1984). [29] L.A. B´aez and P. Clancy, J. Chem. Phys. 101(11), 9837 (1994). [30] B. Taras and A.D.J. Haymet, Molecular Simulation 30, 131 (2003). [31] A. Raman, Phys. Rev. 136, 405 (1964). [32] G. Sutmann and R. Vallauri, J. of Mol. Liq. 98-99, 213 (2002).