上海交通大学python期末大作业题目(姚天昉)

姚天昉是上海交大一位非常有名的计算机老师,期末大作业非常难,给弟弟妹妹们透露下。ps：这是大作业一,另有大作业二即将公布,敬请关注!

Project A

Simulating a Physical System

Introduction

You have already seen a number of examples using randomization to solve problems and to generate experimental results. In the percolation project, grids were generated according to a specified probability distribution to experimentally determine the percolation probability. This approach is an example of a Monte Carlo method, an algorithm that relies on repeated random sampling to compute results. Monte Carlo methods are often used for simulating physical and mathematical systems when other methods are impossible or computationally infeasible. Objective and Background

The objective of this project is to use the Monte Carlo “demon algorithm” to estimate parameters (for example, temperature) in a physical system, specifically an ideal gas with moving molecules. We could simulate such a system by computing the pairwise interactions among all molecules at every time step in the simulation, but that approach would be computationally intractable. Moreover, in these systems we're not typically interested in the specific behavior of each individual molecule, but rather in the overall system behavior. For example, we want to know the expected distribution of molecule velocities in a gas (e.g., to understand its temperature), not which molecules have collided with which other molecules.

The key concept behind the demon algorithm is that any simulation must both preserve the total energy of the system (energy is conserved) and insure that no particle winds up with negative kinetic energy. The algorithm proceeds by choosing a molecule at random and proposing to make a small random change to its energy (e.g., to simulate the effect of a collision). To compensate for the change in energy of the molecule / particle, the opposite change is made to a special “demon” molecule, thus conserving the total energy of the system. Note that the demon molecule is never chosen at random for this change; it only acts to store the difference between the system energy and the total energy.

Before describing the demon algorithm in detail, we give a brief description of the system to be simulated. In the ideal gas, we want to study the velocity of molecules in the gas. This average velocity gives a measure of the temperature of the gas. Consider the state of the gas in an insulated container at an instant of time. Each of its molecules has some velocity that remains constant until the next collision involving the molecules. Because the molecules move quickly, on average, many of them will have experienced collisions during a short time span. Consequently, most of the molecules will have substantially different velocities than they originally did. This process can, in principle, be simulated directly. In practice, doing so is computationally too expensive or is infeasible (e.g., one step of the situation could mean solving n differential equations, with each equation having n² terms).

姚天昉是上海交大一位非常有名的计算机老师,期末大作业非常难,给弟弟妹妹们透露下。ps：这是大作业一,另有大作业二即将公布,敬请关注!

Sketch of Demon Algorithm

The demon algorithm is a common technique for simulating selected parameters of a complex system by maintaining a given macroscopic (total) energy, while randomly changing the microstate of the system. If a trial change would result in a reduction of energy in the system, additional energy is given to a demon (fictitious). If the change would result in additional system energy, the difference is taken from the demon (assuming it has sufficient energy). In our simulation, making a trial change will refer to changing a particle's velocity.

Initialize the system state. The sum of the system energy and the demon energy always equals the specified total energy. The demon energy must always be non-negative.

Execute one simulation step. One simulation step consists of making a specified number of trials, with each trial doing the following: Make a change to the state of the system and record the change in system energy as deltaEnergy. If the change is legitimate, i.e., if the demon energy resulting from making the change is non-negative, the change is accepted and we set

demonEnergy = demonEnergy - deltaEnergy

systemEnergy = systemEnergy + deltaEnergy

Update any parameters to be maintained during the simulation.

Repeat making trials until one simulation step is complete.

After one simulation step, update any additional parameters. Execute as many simulation steps as specified.

It is not obvious that this process can effectively simulate a physical system. In fact, there is no rigorous proof that it does. In practice, however, this algorithm works very well. The first part of the project uses the demon algorithm in an ideal gas simulation.

Ideal Gas Simulation

An ideal gas is a simple, uniform system: Molecules with energy are bouncing around inside a closed container. The internal molecular dynamics is simple – free motion with occasional intermolecular collisions. When using the demon algorithm for a simulation, we view the result of a collision as a random change of the molecule's velocity. Each molecule has a particular momentum, and collisions exchange part of that momentum.

Consider the velocity of a single molecule over time. Each collision can be thought of as a random change in velocity. Since we are interested in the average velocity of all molecules, the exact number and time of the collisions is not important. Rather, what matters is the total change in velocity over the time period we are considering. The demon algorithm takes advantage of this idea. We choose a total energy for the system (view it as the temperature). One step of the simulation consists of a n …… 此处隐藏：5961字，全部文档内容请下载后查看。喜欢就下载吧 ……