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物理代写|量子场论代写Quantum field theory代考|Simulating the Anharmonic Oscillator

Since a CPU only knows numbers, we rescale the dimensionful constants and coordinates in the action (4.39) with powers of the dimensionful lattice constant $\varepsilon$ to arrive at dimensionless lattice quantities $\left(m_{L}, \mu_{L}, \lambda_{L}, q_{L}\right)$ :
$$q_{L}=q / \varepsilon, \quad m_{L}=\varepsilon m, \quad \mu_{L}=\varepsilon^{3} \mu \quad \text { and } \quad \lambda_{L}=\varepsilon^{5} \lambda$$
In terms of these dimensionless quantities, the lattice action takes the simple form
$$S(\boldsymbol{q})=\sum_{j=0}^{n-1}\left{\frac{m_{L}}{2}\left(q_{j+1}-q_{j}\right){L}^{2}+\mu{L} q_{j, L}^{2}+\lambda_{L} q_{j, L}^{4}\right}$$
In a local Metropolis algorithm, we test a new configuration $\boldsymbol{q}^{\prime}$ which differs from the old configuration $q$ only on one lattice point, say lattice point $j$. Then the difference of the two actions is
\begin{aligned} S\left(\boldsymbol{q}^{\prime}\right)-S(\boldsymbol{q}) \approx &\left(q_{j}^{\prime}-q_{j}\right){L}\left{-m{L}\left(q_{j+1}+q_{j-1}\right){L}\right.\ &\left.+\left(q{j}^{\prime}+q_{j}\right){L}\left{m{L}+\mu_{L}+\lambda_{L}\left(q_{j}^{2}+q_{j}^{2}\right)_{L}\right}\right} \end{aligned}
Here the problem arises how to determine energies or lengths in physical units. They are only given in terms of the unknown lattice constant $\varepsilon$ which does not even enter the lattice action $S(\boldsymbol{q})$. Thus one first calculates some observable (e.g., example an energy) which is then compared to the experimentally known value to set the scale.

Alternatively one may express all dimensionful quantities in units of a known and fixed unit of length $\ell$,
$$\varepsilon=a \ell, \quad q=\tilde{q} \ell, \quad m=\tilde{m} / \ell, \quad \mu=\tilde{\mu} / \ell^{3} \quad \text { and } \quad \lambda=\tilde{\lambda} / \ell^{5}$$
In this system of units
\begin{aligned} S\left(\boldsymbol{q}^{\prime}\right)-S(\boldsymbol{q})=&\left(\tilde{q}{j}^{\prime}-\tilde{q}{j}\right)\left{-\frac{\tilde{m}}{a}\left(\tilde{q}{j+1}+\tilde{q}{j-1}\right)\right.\ &\left.+\left(\tilde{q}{j}^{\prime}+\tilde{q}{j}\right)\left{\frac{\tilde{m}}{a}+a \tilde{\mu}+a \tilde{\lambda}\left(\tilde{q}{j}^{n 2}+\tilde{q}{j}^{2}\right)\right}\right} \end{aligned}
This formula is used in the header file stdanho. h on page 80 which is called by the program anharmonic1.c listed on p. 77. The parameters $m, \mu, \lambda$, the number of lattice points (in the C-program $n$ is renamed $N$ ), and the lattice constant $a$ are all defined in another header file constants. hon p. 79. In order to have uncorrelated configurations, only one out of $M A$ configuration is measured. In addition, since a Markov chain needs some time to reach equilibrium, we start measuring configurations only after $M A \times M E$ sweeps through the lattice.

To measure the square of $\psi_{0}(q)$ on the interval $[-I N T E R V, I N T E R V]$ with the help of (4.49), we divide the interval into BIN bins. With the parameter DELTA, we adjust the amplitude of a tentative coordinate change during a local update according to $q^{\prime}=q+D E L T A \times(1-2 r)$, where $r$ is a uniform random number on $[0,1]$. The program anharmonic $1 .$. generates the histogram of the probability distribution. In Fig. $4.4$ we plotted the so obtained density $\left|\psi_{0}(q)\right|^{2}$, both for the harmonic and the anharmonic oscillator.

物理代写|量子场论代写Quantum field theory代考|Hybrid Monte Carlo Algorithm

The powerful hybrid Monte Carlo algorithm has been developed by S. DUANE ET AL. [10]. A recommendable introduction can be found in [11], and in our presentation, we follow in part the reviews $[12,13]$. The algorithm represents a combination of molecular dynamics (see [14]) and Metropolis algorithm. It aims at global updates of whole configurations with reasonable large acceptance rates, in order to minimize the time required to generate independent configurations.

Molecular dynamics (MD) simulations are frequently and successfully used to study (classical) many-body systems and are applied to problems in material science, astrophysics, and biophysics. In molecular dynamics simulations, one solves the Hamiltonian cquations of motion
$$\dot{q}{i}=\frac{\partial H}{\partial p{i}} \quad, \quad \dot{p}{i}=-\frac{\partial H}{\partial q{i}}$$
numerically and makes use of the ergodic hypothesis, according to which the statistical ensemble averages are equal to time averages. From an initial configuration, represented by a point $\left(\boldsymbol{q}{0}, \boldsymbol{p}{0}\right)$ in phase space, one obtains a unique solution of Hamilton’s equations of motion. Without any numerical errors, the energy is a constant of motion, and this simple observation will be relevant in what follows.
In Euclidean quantum mechanics discretized on an imaginary-time lattice $\Lambda$ with $n$ points ${x}$, we introduce an extended phase space with dimension $2 n$. Each point in extended phase space consists of a broken-line path characterized by variables $\left{q_{x}\right}$ and their canonically conjugated momenta $\left{p_{x}\right}$. To construct a Markov process with global updates and high acceptance rate, we introduce the following auxiliary Hamiltonian in extended phase space:
$$H(\boldsymbol{q}, \boldsymbol{p})=\frac{\boldsymbol{p}^{2}}{2}+S(\boldsymbol{q}), \quad \boldsymbol{p}^{2}=\sum_{x \in \Lambda} p_{x}^{2}$$

物理代写|量子场论代写Quantum field theory代考|Implementing the HMC Algorithm

To fulfil the condition of detailed balance, we need a time-reversible and symplectic integrator to numerically solve the fictitious Hamiltonian dynamics. Using the naive forward difference operator $\dot{f}(\tau) h \approx f(\tau+h)-f(\tau)$ in the discretized equations of motion leads to the time-irreversible prescription
\begin{aligned} &\boldsymbol{q}(\tau+h)=\boldsymbol{q}(\tau)+h \boldsymbol{p}(\tau) \ &\boldsymbol{p}(\tau+h)=\boldsymbol{p}(\tau)+h \boldsymbol{F}(\boldsymbol{q}(\tau)) \end{aligned}
with force $\boldsymbol{F}=-\nabla_{q} S$. This prescription must not be used in any HMC algorithm. In most simulation the time-reversible leapfrog integration is used instead. Here one first moves forward with the momenta a half step in fictitious time. Then one moves forward several time-steps alternately with coordinates $\boldsymbol{q}$ and momenta $\boldsymbol{p}$. The last move of the momenta is again a half step in fictitious time.

物理代写|量子场论代写Quantum field theory代考|Simulating the Anharmonic Oscillator

q大号=q/e,米大号=e米,μ大号=e3μ 和 λ大号=e5λ

\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别

\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别

e=一个ℓ,q=q~ℓ,米=米~/ℓ,μ=μ~/ℓ3 和 λ=λ~/ℓ5

\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别

物理代写|量子场论代写Quantum field theory代考|Hybrid Monte Carlo Algorithm

S. DUANE 等人开发了强大的混合蒙特卡罗算法。[10]。推荐的介绍可以在 [11] 中找到，在我们的演示文稿中，我们部分遵循了评论[12,13]. 该算法代表了分子动力学（参见 [14]）和 Metropolis 算法的组合。它旨在以合理的大接受率对整个配置进行全局更新，以最大限度地减少生成独立配置所需的时间。

q˙一世=∂H∂p一世,p˙一世=−∂H∂q一世

H(q,p)=p22+小号(q),p2=∑X∈ΛpX2

物理代写|量子场论代写Quantum field theory代考|Implementing the HMC Algorithm

q(τ+H)=q(τ)+Hp(τ) p(τ+H)=p(τ)+HF(q(τ))