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leukos

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[求助] 求问分子模拟中提高温度是否就能实现增强抽样呢？

http://wenku.baidu.com/link?url= ... OWPEiaMSpcSOGRSvStm
最近在看上面这篇文章，想对分子模拟中的增强抽样方法有一个基本的了解
————————————————

想问一个问题
提高温度是不是就能达到增强抽样的目的了呢？
比如原来的温度是T1
现在把温度变成更高的温度T2，在这个温度下，就能达到增强抽样了？
怎么理解呢？

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1楼 2014-03-28 13:46:45

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lsloneil

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leukos: 金币+20, ★★★很有帮助 2014-04-07 19:45:42

是的，升温可以增强抽样。因为温度对应于分子的平均动能，当势能面上有一个很深的势阱的时候在里面的粒子由于动能不够高是很难跳出这个势阱的，这样的话你的采样基本就在势阱里面采样而无法在势能面的其他位置采样，升高温度就可以使粒子跳出势阱。升温来增加抽样的经典方法是Replica Exchange MD (REMD).

Enhanced Sampling是分子模拟中当前非常热门的课题。你可以看看本版EPI ChemiAndy写的这篇关于自由能计算的文章：http://muchong.com/bbs/viewthread.php?tid=3672663
里面谈到的Umbrella Sampling, WHAM等都是常用的方法。

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3楼2014-04-07 02:00:43

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leukos: 金币+45, ★★★★★最佳答案, 再次感谢～～思路清晰了不少呢 2014-04-08 12:48:00
月只蓝: 金币+10, 感谢您的耐心和劳动付出！ 2014-04-08 19:03:13

引用回帖:

4楼: Originally posted by leukos at 2014-04-07 01:13:09
另外希望问问上面这张图
来自 Chemical Physics Letters 332 (2000) 131-138

发现还是不是很能理解 Probability distribution of potential energy的意义
是指模拟过程中，分子在某个特定温度下，处于势 ...

见下，我在latex里用英文敲的，你凑合着看吧。

\section{First Question}
I think your answer with regard to the probability distribution of potential energy is correct. There is indeed an expression
in term of ensemble average for this probability distribution, which is given below.

For a system with N particles, the potential energy is ${V_N(\textbf{r}^N)}$ , so the configuration partition function is

${ Z_N = \int d\textbf{r}^N e^{-\beta V_N(\textbf{r}^N)}}$ ,

where ${\beta = 1/k_BT}$ . So for an ensemble average of function ${f(\textbf{r}^N)}$ is given by

${\langle f \rangle = \int d\textbf{r}^N f(\textbf{r}^N) e^{-\beta V_N(\textbf{r}^N)}/Z_N.}$

Then the probability distribution of potential energy P(U) can be written as

${ P(U)\equiv \langle \delta (V_N(\textbf{r}^N) - U) \rangle = \int d\textbf{r}^N \delta (V_N(\textbf{r}^N) - U) e^{-\beta V_N(\textbf{r}^N)}/Z_N.}$

For most systems, there is no analytical way of calculating P(U), but one could just sample P(U) in simulations,
which can be done very easily. In general the P(U) obtained from sampling is not perfectly Gaussian, but one could fit
it to Gaussian.

\section{Second Question}
From the figure you showed, one could see that the mean value of potential energy increases as temperature increases.
Also, the width of the potential energy distribution (i.e. fluctuations) also increases as temperature increases. This is a natural
result from thermodynamics and statistical mechanics. First, the derivative of mean potential energy with respect to temperature
is just the heat capacity (here it means the non-ideal part of the heat capacity).

${\frac{\partial \langle V_N \rangle}{\partial T}=Nc_v.}$

Apparently the heat capacity is greater than zero (because of the convexity of free energy function), so the mean potential
energy increases with temperature.

On the other hand, one could easily find the relation between fluctuation of energy (the width of potential energy distribution)
and temperature in a standard stat mech textbook.

${\langle V_N - \langle V_N \rangle \rangle = Nk_BT^2 c_v,}$

So the fluctuation increases as temperature increases.

So, to answer your question, I would say that in canonical (NVT) ensemble, molecules would always prefer configurations
that minimizes free energy. Since free energy comprises parts both from energy and entropy, molecules have to choose
configurations that find a balance between energy and entropy (i.e. minimize energy but also maximize entropy).

\section{Last Question}
The last question is much easier to answer. Suppose to get sufficient statistics, you need to sample at least 100 ps in
the region of stable configuration. However, since you're using REMD, you spend 50 ps of your 100 ps exploring other
parts of the phase space, then you have to spend more time totally in order to obtain as good statistics as your previous
simulations for your stable configuration.

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6楼2014-04-08 10:36:30

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lsloneil

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月只蓝: 金币+2, 鼓励交流！ 2014-04-10 18:46:30

引用回帖:

7楼: Originally posted by leukos at 2014-04-07 18:26:51
另外，是否可以从中心极限定理来理解那个高斯分布的呢？
但具体我又无法把中心极限定理和这个高斯分布关系起来
两者有关系吗？再次感谢...

呵呵，这个问题我没仔细想过，但据我所知有些势能分布不是高斯分布的，尤其是势能值比较大的一些情况，至于为什么我不太清楚。

至于中心极限定律，它所说的高斯分布大概是这个意思。假设势能在U1处的概率为P1，你对势能进行N次采样，当N趋近于无穷大时，出现势能为U1的次数应该是一个均值为N*P1的高斯分布。

我讲的可能不对，欢迎指正。

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8楼2014-04-09 12:56:59

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leukos

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继续求问...><

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2楼2014-04-06 19:42:30

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leukos

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★ ★ ★
月只蓝: 金币+3, 鼓励交流！ 2014-04-08 19:01:52

引用回帖:

3楼: Originally posted by lsloneil at 2014-04-07 02:00:43
是的，升温可以增强抽样。因为温度对应于分子的平均动能，当势能面上有一个很深的势阱的时候在里面的粒子由于动能不够高是很难跳出这个势阱的，这样的话你的采样基本就在势阱里面采样而无法在势能面的其他位置采样， ...

另外希望问问上面这张图
来自 Chemical Physics Letters 332 (2000) 131-138

发现还是不是很能理解 Probability distribution of potential energy的意义
是指模拟过程中，分子在某个特定温度下，处于势能U 的概率？
可是是否有具体有解析的表达式可以算出这样的一个高斯的分布的样子呢？
还是这种图只能通过模拟来得到呢？

从物理上应该怎样来理解这个图像呢？
比如低温的时候，分子动能小，所以就只能主要处于一个低的势能U上？
不过不是无论温度高低，分子都应该最主要会处在一个能量极小值的位置吗？

感觉还是有点晕。。。求指点呢>< 感谢
————————————————————————————

另外一个问题是：
一楼处的那篇文章中提到，“然而仅仅通过降低势垒或升高温度来增加对小概率事件的取样，虽然可以提高对化学反应或分子构象变化过
程的计算，但由于损失掉了对目标温度下稳定构象
的充分取样，往往会使得模拟中的热力学计算收敛
变慢。”

这个损失掉对目标温度下稳定构象的充分取样是什么意思呢？
没理解这句话。。。。

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4楼2014-04-07 21:13:09

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lsloneil

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$\section{First Question} I think your answer with regard to the probability distribution of potential energy is correct. There is indeed an expression in term of ensemble average for this probability distribution, which is given below. For a system with $N$ particles, the potential energy is $V_N(\textbf{r}^N)$, so the configuration partition function is \begin{equation} Z_N = \int d\textbf{r}^N e^{-\beta V_N(\textbf{r}^N)}, \end{equation} where $\beta = 1/k_BT$. So for an ensemble average of function $f(\textbf{r}^N)$ is given by \begin{equation} \langle f \rangle = \int d\textbf{r}^N f(\textbf{r}^N) e^{-\beta V_N(\textbf{r}^N)}/Z_N. \end{equation} Then the probability distribution of potential energy $P(U)$ can be written as \begin{equation} P(U)\equiv \langle \delta (V_N(\textbf{r}^N) - U) \rangle = \int d\textbf{r}^N \delta (V_N(\textbf{r}^N) - U) e^{-\beta V_N(\textbf{r}^N)}/Z_N. \end{equation} For most systems, there is no analytical way of calculating $P(U)$, but one could just sample $P(U)$ in simulations, which can be done very easily. In general the $P(U)$ obtained from sampling is not perfectly Gaussian, but one could fit it to Gaussian. \section{Second Question} From the figure you showed, one could see that the mean value of potential energy increases as temperature increases. Also, the width of the potential energy distribution (i.e. fluctuations) also increases as temperature increases. This is a natural result from thermodynamics and statistical mechanics. First, the derivative of mean potential energy with respect to temperature is just the heat capacity (here it means the non-ideal part of the heat capacity). \begin{equation} \frac{\partial \langle V_N \rangle}{\partial T}=Nc_v. \end{equation} Apparently the heat capacity is greater than zero (because of the convexity of free energy function), so the mean potential energy increases with temperature. On the other hand, one could easily find the relation between fluctuation of energy (the width of potential energy distribution) and temperature in a standard stat mech textbook. \begin{equation} \langle V_N - \langle V_N \rangle \rangle = Nk_BT^2 c_v, \end{equation} So the fluctuation increases as temperature increases. So, to answer your question, I would say that in canonical (NVT) ensemble, molecules would always prefer configurations that minimizes free energy. Since free energy comprises parts both from energy and entropy, molecules have to choose configurations that find a balance between energy and entropy (i.e. minimize energy but also maximize entropy). \section{Last Question} The last question is much easier to answer. Suppose to get sufficient statistics, you need to sample at least 100 ps in the region of stable configuration. However, since you're using REMD, you spend 50 ps of your 100 ps exploring other parts of the phase space, then you have to spend more time totally in order to obtain as good statistics as your previous simulations for your stable configuration. $

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5楼2014-04-08 10:23:11

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leukos

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6楼: Originally posted by lsloneil at 2014-04-08 10:36:30
见下，我在latex里用英文敲的，你凑合着看吧。

\section{First Question}
I think your answer with regard to the probability distribution of potential energy is correct. There is indeed an expressi ...

另外，是否可以从中心极限定理来理解那个高斯分布的呢？
但具体我又无法把中心极限定理和这个高斯分布关系起来
两者有关系吗？再次感谢

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7楼2014-04-08 14:26:51

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leukos

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月只蓝: 金币+1, 鼓励交流！ 2014-04-15 18:25:54

引用回帖:

8楼: Originally posted by lsloneil at 2014-04-09 12:56:59
呵呵，这个问题我没仔细想过，但据我所知有些势能分布不是高斯分布的，尤其是势能值比较大的一些情况，至于为什么我不太清楚。

至于中心极限定律，它所说的高斯分布大概是这个意思。假设势能在U1处的概率为P1， ...

谢谢哈~
另外
http://en.wikipedia.org/wiki/Thermal_fluctuations
貌似wiki这边提到了一些，我还没去仔细理解
晚点有空我再看看

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9楼2014-04-13 18:15:28

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