A Brownian particle of mass \(m\) is attached to a harmonic spring with force constant, \(k\), and is driven by an external force, \(F(t)\). The particle is constrained to move in one dimension. The Langevin equation is $$ m \frac{\mathrm{d}^{2} x(t)}{\mathrm{d} t^{2}}+\gamma \frac{\mathrm{d} x(t)}{\mathrm{d} t}+m \omega_{0}^{2} x(t)=\xi(t)+F(t) $$ where \(\omega_{0}=k / m, \gamma\) is the friction constant, and \(\mathcal{\xi}(t)\) is a Gaussian white noise with zero mean, \((\xi(t)\rangle_{\varepsilon}=0 .\) Here ()\(\varepsilon\) denotes the average over values of the random force. Consider the overdamped case. (a) Compute the equilibrium correlation function, \((\langle x(t) x(0)\rangle \xi\rangle T\), starting from the Langevin equation above with \(F(t)=0\). Let (\rangle\(_{T}\) denote the thermal average over the initial position and velocity of the Brownian particle. Assume that \((x(0) v(0)\rangle_{T}=0\) and \(\left(x(0)^{2}\right\rangle_{T}=k_{\mathrm{B}} T /\left(m \omega_{0}^{2}\right)\), (b) Show that the dynamic susceptibility for the Brownian oscillator is \(X(\omega)=\left(-m \omega^{2}+m \omega_{n}^{2}-\mathrm{i} \gamma \omega\right)^{-1}\). Use the fluctuation- dissipation theorem, $$ (x(t) x(0)\rangle=\frac{k_{\mathrm{B}} T}{\mathrm{i} \pi} P \int_{-\infty}^{\infty} \frac{1}{\omega} X(\omega) \cos (\omega t) $$ to compute the equilibrium correlation function. Do your results in (a) and (b) agree?

Step by step solution

01

- Langevin Equation Simplification

Start with the Langevin equation: \[ m \frac{\text{d}^{2} x(t)}{\text{d} t^{2}} + \beta \frac{\text{d}x(t)}{\text{d} t} + k x(t) = \tilde{\beta}(t) \] Given that $$ F(t) = 0 $$, and for the Brownian Motion case we have that the particle is in overdamped condition where $$ \beta > 1 $$. Thus $$ m \frac{\text{d}^{2}x(t)}{\text{d}t^2}=0 $$. Thus we reduce the equation to: \[ \beta \frac{\text{d}x(t)}{\text{d}t} + k x(t) =\tilde{\beta}(t) \] We are only interested in solving the steady-state condition where $$ \tilde{\beta}=0$$ thus reducing the equation will become: \[ \beta \frac{\text{d}x(t)}{\text{d}t} + k x(t) = 0 \]

02

- Solving for Steady-state

Apply the initial condition given: \[ \beta \frac{\text{d}x(0)}{\text{d}t} + k x(0) = 0 \] Let $$ \frac{\text{d}x}{\text{d}t}=v(t) $$ and solve: $$v(t)+\frac{k}{\beta} x(t) = 0$$

03

- Solving for x(t)

Separate variables and integrate: $$ dx/x = -(\beta/k) dt $$ and integrate each side to give us: $$ \text{ln}(x(t)/A)=\beta(-t)\$$ Take both sides to the power of e, We recover an expression for position x(t) = A*(exp-t*k/\beta)

04

- Calculate Correlation Function

We assume the given properties for Brownian system: $$ x(0)=k(\frac{T_b}{k m}$$ Given we have: $$ V(0)=0 \ \text{Now apply this property and compute the Fluctuations-Dissipation theorem:} (x(t)x(0))\rangle=\frac {k_{\tilde{B}}} {T} P\int_{- infinity}^{infinity}\frac{1}{w} k(\right)cos(\right t)\rangle_{\beta} Apply this theorem to $(gamma=\r(x(t)}=v)_r=g(vd) \ (results a

05

- Dynamic Susceptibility

The Dynamic susceptibility =$$X(\tilde{\beta})q=k(\beta/k*p)=-m(\beta)+result m(\beta^2)-mr(\beta^1q)\ use <-form and denote as = says a coherence on oscillator

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Langevin equation

The Langevin equation is a differential equation that describes the motion of a particle in a fluid, accounting for both systematic forces and random forces. It is written as:

\[m \frac{\mathrm{d}^{2} x(t)}{\mathrm{d} t^{2}} + \gamma \frac{\mathrm{d}x(t)}{\mathrm{d} t} + m \omega_{0}^{2} x(t) = \xi(t) + F(t)\]

Here,

• \(m\) is the mass of the particle
• \(\gamma\) is the friction constant
• \(\omega_{0}^{2}\) is the harmonic oscillator's natural frequency squared
• \(\xi(t)\) is a Gaussian white noise term representing random forces
• \(F(t)\) is any external force acting on the particle

In an overdamped system where the friction is strong, the term with \(m \frac{\mathrm{d}^{2} x(t)}{\mathrm{d} t^{2}}\) can be neglected.
Thus, the equation simplifies to:

\[\gamma \frac{\mathrm{d}x(t)}{\mathrm{d} t} + k x(t) = \xi(t)\]

This helps us understand the balance between damping forces and random thermal movements in the system.

Harmonic oscillator

A harmonic oscillator is a system that experiences a restoring force proportional to its displacement from an equilibrium position. The force can be described by Hooke's Law as:

\[F = -kx\]

Here,

• \(F\) is the restoring force
• \(k\) is the force constant or stiffness
• \(x\) is the displacement

In our context, the particle in the Langevin equation is attached to a harmonic spring. This attachment results in the term \(m \omega_{0}^{2} x(t)\) in the Langevin equation, where \(\omega_{0} = \sqrt{k/m}\).
This term encapsulates the harmonic behavior of the system.

Fluctuation-dissipation theorem

The fluctuation-dissipation theorem links the random fluctuations in a system (represented by \(\xi(t)\)) with its response to external forces (as described by dynamic susceptibility).
It is one of the central principles in statistical mechanics and can be expressed as:

\[\langle x(t) x(0) \rangle = \frac{k_{B} T}{\text{i} \pi} P \int_{-\infty}^{\infty} \frac{1}{\omega} X(\omega) \cos (\omega t) d\omega\]

Here,

• \(\langle x(t) x(0) \rangle\) is the equilibrium correlation function
• \(k_{B} T\) is the thermal energy
• \(X(\omega)\) is the dynamic susceptibility

This theorem helps in calculating the equilibrium correlation function by integrating over the system's response spectrum.

Equilibrium correlation function

The equilibrium correlation function \(\langle x(t) x(0) \rangle\) measures the statistical dependence of the particle’s position at different times.

For our system, this function can be computed from the Langevin equation assuming \(F(t) = 0\), giving us insight into how the position of the particle at time \(t\) relates to its position at time zero.

The computation starts with solving the simplified Langevin equation:

\[ \gamma \frac{\mathrm{d}x(t)}{\mathrm{d} t} + k x(t) = \xi(t)\]

After integrating and applying initial conditions, you can relate the solution to thermal averages, yielding:

\[ \langle x(t)x(0) \rangle = \frac{k_{B} T}{k} e^{-\frac{k}{\gamma} t}\]

This result shows that the correlation decays exponentially over time with a rate determined by the damping and spring constants.

Dynamic susceptibility

Dynamic susceptibility, denoted as \(X(\omega)\), describes how the system responds to an external periodic force with frequency \(\omega\). It is particularly important for understanding the system's behavior in different frequency regimes.

For a Brownian oscillator, dynamic susceptibility is given by:

\[ X(\omega) = \left(-m \omega^{2} + m \omega_{0}^{2} - \text{i} \gamma \omega\right)^{-1} \]

This function outlines how the system reacts dynamically to external forces. For example:

• At low frequencies, the system can follow the applied force more easily.
• At high frequencies, inertia and damping makes it harder for the system to respond.

Utilizing this function alongside the fluctuation-dissipation theorem, we can compute other relevant properties like the equilibrium correlation function.