Histogram potential

Fig. 4. A, Histogram of single channel conductances, y, of Gramicidin A in diphytanoyl L-a-lecithin/n-decane membranes for 1 M KC1, 130 mV potential and 40 °C. Note that single channel conductance is simply the single channel current, i , divided by the applied potential. Reproduced from reference 12) with permission.

B, Histogram of channel lifetimes for Gramicidin A in diphytanoyl L-a-lecithin/n-decane membranes for 1 M KC1, 130 mV applied potential and 15 °C. 

C. Histogram of single channel conductances of Gramicidin A in glyceryl monoolein/hexadecane membranes for 1 M KC1, 103 mV applied potential and 23 °C. Reproduced with permission from Ref. 13>... 

For the analysis, we developed a new method that makes it possible to observe correlated potentials between two trapped particles. The principle is shown in Figure 7.5. From the recorded position fluctuations of individual particles (indicated by the subscripts 1 and 2), histograms are obtained as a function of the three-dimensional position. Since the particle motion is caused by thermal energy, the three-dimensional potential proflle can be determined from the position histogram by a simple logarithmic transformation of the Boltzmarm distribution. Similarly, the... 

A brief sketch of the derivation of the WHAM equations follows we note that a detailed explanation is available in the book by Frenkel and Smit [14], Consider the canonical reweighing (3.5). Our goal will be to combine the histograms pi(U) from several runs at different temperatures T to predict the distribution of potential energies at a new temperature T. Individually, each run enables us to reweight its histogram to obtain the distribution at T... 

In contrast to the weights formalism, the partition function approach directly employs the ideal flat-histogram expression in (3.36). Its goal is not to determine q but Q(N. V, T) directly, or more precisely in this case, the N dependence of Q. Due to numerical reasons, we usually work instead with the associated thermodynamic potential which is the logarithm of the partition function of interest in this case it is In Q = — 7/1 =. A, where we have used script i as an abbreviation. Thus our sampling scheme becomes... 

Not surprisingly, the essential component of flat-histogram algorithms is the determination of the weights, r/, or the thermodynamic potential, e.g., / or /. There exist a number of techniques for accomplishing this task. The remainder of this section is dedicated to reviewing a small but instructive subset of these methods, the multicanonical, Wang-Landau, and transition-matrix approaches. We subsequently discuss their common and sometimes subtle implementation issues, which become of practical importance in any simulation. 

Fig. 3.4. Evolution of the weights i],(N) and the histograms fi(N) in a grand-canonical implementation of the multicanonical method for the Lennard-lones fluid al V 125. The temperature is T = 1.2 and the initial chemical potential is /./ = —3.7. The weights are updated after each 10-million-step interval, and the numbers indicate the iteration number. The second peak in the weights at large particle numbers indicates that the initial chemical potential is close to its value at coexistence...

Fig. 8.3. Histogram of work values for Jarzynski s identity applied to the double-well potential, V(x) = x2(x — a)2 + x, with harmonic guide Vpun(x, t) = k(x — vt)2/2, pulled with velocity v. Using skewed momenta, we can alter the work distribution to include more low-work trajectories. Langevin dynamics on Vtot(x(t),t) = V(x(t)) + Upuii(x(t)yt) with JcbT = 1, k = 100, was run with step size At = 0.001, and friction constant 7 = 0.2 (in arbitrary units). We choose v = 4 and a = 4, so that the barrier height is many times feT and the pulling speed far from reversible. Trajectories were run for a duration t = 1000. Work histograms for 10,000 trajectories, for both equilibrium (Maxwell) initial momenta, with zero average and unit variance, and a skewed distribution with zero average and a variance of 16.0...

In summary, the Gibbs ensemble MC methodology provides a direct and efficient route to the phase coexistence properties of fluids, for calculations of moderate accuracy. The method has become a standard tool for the simulation community, as evidenced by the large number of applications using the method. Histogram reweighting techniques (Chap. 3) have the potential for higher accuracy, especially if... 

Histograms for two runs at different chemical potentials are presented in Fig. 10.2. There is a range of N over which the two runs overlap. In Fig. 10.3 we show the function lnp(iV) — f S/iN for the data of Fig. 10.2. Rearranging (10.15) and taking the logarithm, we see that this function is related to the Helmholtz free energy... 

The histogram reweighting methodology for multicomponent systems [52-54] closely follows the one-component version described above. The probability distribution function for observing Ni particles of component 1 and No particles of component 2 with configurational energy in the vicinity of E for a GCMC simulation at imposed chemical potentials /. i and //,2, respectively, at inverse temperature ft in a box of volume V is... 

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