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Wednesday, November 25

Thursday, November 19

  1. page QMC for harmonic bosons edited ... W_{\geq k}= Z_{N-k} -2- Consider the partition function Wk of N bosons with precisely k of th…
    ...
    W_{\geq k}= Z_{N-k}
    -2- Consider the partition function Wk of N bosons with precisely k of them in the ground state. Show in details that
    W_{k}= \left\{\begin{array}{ll}W_{\geq\quad \quad W_{k}= \left\{\begin{array}{ll}
    W_{\geq
    k}-W_{\geq k+1} & \text{if}\quad k<n\\ w_{\geq= k}&= \text{if}\quad=\text{if} \quad k< n \\
    W_{\geq k} & \text{if} \quad
    k= Nn \end{array} \right.
    w_{\geq= k}&= \text{if}\quad= k=N

    -3- Deduce from those results the probability π(N0) of having N0 bosons in the ground state in terms of the partition function.
    -4- The condensate fraction, which is the mean value ⟨N0⟩ of the number N0 of bosons in the ground state, writes
    ...
    Landsberg P. T. (1961) Thermodynamics with quantum statistical illustrations, Interscience Publishers
    SMAC part 4.2.3 - 4.2.6
    [Print<n\\ w_{\geq="k}&=" \text{if}\quad="k=" n="" \end{array}="" \right.<br=""> [Print this page]
    (view changes)
  2. page QMC for harmonic bosons edited ... W_{\geq k}= Z_{N-k} -2- Consider the partition function Wk of N bosons with precisely k of th…
    ...
    W_{\geq k}= Z_{N-k}
    -2- Consider the partition function Wk of N bosons with precisely k of them in the ground state. Show in details that
    ...
    \text{if}\quad k<n\\ w_{\geq="" k}&="" \text{if}\quad="" k="N\end{array}\right.<br">[[math]]
    <n\\ w_{\geq="" k}&="" \text{if}\quad="" k="N\end{array}\right.<br"></span>
    * -3-
    w_{\geq= k}&= \text{if}\quad= k= N \end{array} \right.
    w_{\geq= k}&= \text{if}\quad= k=N
    -3-
    Deduce from
    ...
    the probability //π//(//N//<span style="font-size: 80%; vertical-align: sub;">0</span>)π(N0) of having //N//<span style="font-size: 80%; vertical-align: sub;">0</span>N0 bosons in
    ...
    partition function.
    * -4-

    -4-
    The condensate
    ...
    mean value ⟨//N//<span style="font-size: 80%; vertical-align: sub;">0</span>⟩⟨N0⟩ of the number //N//<span style="font-size: 80%; vertical-align: sub;">0</span>N0 of bosons
    ...
    state, writes
    <span style="display: block; text-align: center;">

    \langle N_0\rangle = \sum_{N_0=0}^N N_0 \pi(N_0)
    </span>
    > Using
    Using the results
    ...
    show that
    <span style="display: block; text-align: center;">

    \langle N_0\rangle = \frac 1{Z_N}\sum_{p=0}^{N-1} Z_p
    math
    -5- Modify the program you've written in the previous section so as to include a computation of ⟨N0⟩. Plot this quantity as a function of the reduced temperature T٭ for different increasing values of N. Comment.
    References
    (view changes)
  3. page QMC for harmonic bosons edited ... W_{\geq k}= Z_{N-k} -2- Consider the partition function Wk of N bosons with precisely k of th…
    ...
    W_{\geq k}= Z_{N-k}
    -2- Consider the partition function Wk of N bosons with precisely k of them in the ground state. Show in details that
    ...
    & \text{if}\quad k<N\\ W_{\geq k}& \text{if}\quad k=N\end{array}\right.
    -3-
    k<n\\ w_{\geq="" k}&="" \text{if}\quad="" k="N\end{array}\right.<br">[[math]]
    <n\\ w_{\geq="" k}&="" \text{if}\quad="" k="N\end{array}\right.<br"></span>
    * -3-
    Deduce from
    ...
    the probability π(N0)//π//(//N//<span style="font-size: 80%; vertical-align: sub;">0</span>) of having N0//N//<span style="font-size: 80%; vertical-align: sub;">0</span> bosons in
    ...
    partition function.
    -4-

    * -4-
    The condensate
    ...
    mean value ⟨N0⟩⟨//N//<span style="font-size: 80%; vertical-align: sub;">0</span>⟩ of the number N0//N//<span style="font-size: 80%; vertical-align: sub;">0</span> of bosons
    ...
    state, writes
    <span style="display: block; text-align: center;">

    \langle N_0\rangle = \sum_{N_0=0}^N N_0 \pi(N_0)
    Using</span>
    > Using
    the results
    ...
    show that
    <span style="display: block; text-align: center;">

    \langle N_0\rangle = \frac 1{Z_N}\sum_{p=0}^{N-1} Z_p
    math
    -5- Modify the program you've written in the previous section so as to include a computation of ⟨N0⟩. Plot this quantity as a function of the reduced temperature T٭ for different increasing values of N. Comment.
    References
    (view changes)

Wednesday, September 16

  1. page Molecular dynamics - symplectic integrators and event driven dynamics edited ... Event driven dynamics was invented by Alder and Wainwright, in 1957. Here is a cartoon of the …
    ...
    Event driven dynamics was invented by Alder and Wainwright, in 1957. Here is a cartoon of the time-evolution of four hard disks in a square box.
    {Event_movie.jpg} Event-driven Molecular Dynamics simulation for 4 disks in a box
    ...
    with a wall (see ).Inwall. In this simulation,
    ...
    on the lefttop is SMAC
    ...
    test ergodicity? Explain the movie
    {Event_chain_box.gif} Molecular Dynamics evolution for four hard disks in a box with walls (simulation by Maxim Berman).
    Wall collisions, pair collisions
    Wall collisions are trivial. Pair collisions (with periodic boundary conditions) will be treated and programmed in this week's practical session.

    Chaos
    {Event_movie_single_double.jpg} The same MD simulation executed with different precisions of arithmeticThe event-driven molecular dynamics algorithm has no time-step error, and the only source of error comes from the finite precision of the arithmetic. These errors are magnified from iteration to iteration. The cartoons on the left illustrates the influence of tiny rounding errors on the dynamics. This is SMAC fig. 2.5. This extreme influence on the initial conditions is called ''chaos''. In this simulation, chaos has two consequences:
    (view changes)
    10:11 am
  2. page Molecular dynamics - symplectic integrators and event driven dynamics edited ... \mathcal H_{\text{pendulum}} = \frac{p^2}{2} - \cos q Symplectic integrators p_{n+1} = p_n …
    ...
    \mathcal H_{\text{pendulum}} = \frac{p^2}{2} - \cos q
    Symplectic integrators
    p_{n+1} = p_np_
    {t+\Delta t}
    = p_t
    - \Delta
    \left(
    ...
    H}{\partial q} \right)_{q=q_n, p=p_{n+1}}\right)_{q=q_t, p=p_
    {t+\Delta t}
    }
    \quad
    q_{n+1} = q_n

    q_
    {t+\Delta t}
    = q_t
    + \Delta
    \left(
    ...
    H}{\partial p} \right)_{q=q_n, p=p_{n+1}}\right)_{q=q_t, p=p_
    {t+\Delta t}
    }
    \quad
    Part II: The ergodic hypothesis
    Part III: The hard-sphere model
    Short history:
    The study of hard-sphere systems goes back a long time. Many people see a precursor in the Roman poet and philosopher Lucretius. Another mile-stone was Daniel Bernoulli's (1700 – 1782) discussion of the pressure dependence of a hard-sphere gas (1738). Boltzmann worked on hard spheres, so did Maxwell, and many researchers since then. The phase transition in hard disks, discovered by Alder and Wainwright in 1962 is an important discovery made by numerical simulations using the event-driven algorithm.
    Animation
    {Event_chain_box.gif} Molecular Dynamics evolution for
    Event driven dynamics was invented by Alder and Wainwright, in 1957. Here is a cartoon of the time-evolution of four hard
    ...
    in a box with walls (simulation by Maxim Berman).
    Scheme
    square box.
    {Event_movie.jpg} Event-driven Molecular Dynamics simulation for 4 disks in a box
    Here is a cartoon of the time-evolution of four hard disks in a square box. AtAt time t=0,
    ...
    wall (see ).
    In
    ).In this simulation,
    ...
    fig. 2.1.
    How to test ergodicity?
    {Event_chain_box.gif}
    Molecular dynamics was invented by Alder and Wainwright,Dynamics evolution for four hard disks in 1957.a box with walls (simulation by Maxim Berman).
    Wall collisions, pair collisions
    Wall collisions are trivial. Pair collisions (with periodic boundary conditions) will be treated and programmed in this week's practical session.
    (view changes)
    9:48 am
  3. page Molecular dynamics - symplectic integrators and event driven dynamics edited ... is a constant of the motion. Euler Discretization ... t} = p_n p_t - \Delta q_{t+\Delt…
    ...
    is a constant of the motion.
    Euler Discretization
    ...
    t} = p_np_t - \Delta
    q_{t+\Delta t}
    = q_t + \Delta t
    (view changes)
    4:55 am
  4. page Molecular dynamics - symplectic integrators and event driven dynamics edited ... is a constant of the motion. Euler Discretization p_{n+1} p_{t+\Delta t} = p_n ... H}{\p…
    ...
    is a constant of the motion.
    Euler Discretization
    p_{n+1}p_{t+\Delta t} = p_n
    ...
    H}{\partial q} \right)_{q=q_n, p=p_n}\right)_{q=q_t, p=p_t} \quad
    q_{n+1} = q_n

    q_{t+\Delta t}
    = q_t
    + \Delta
    \left(
    ...
    H}{\partial p} \right)_{q=q_n, p=p_n}\right)_{q=q_t, p=p_t} \quad
    import pylab
    Ntime = 200
    (view changes)
    4:55 am
  5. page Molecular dynamics - symplectic integrators and event driven dynamics edited ... Part III: The hard-sphere model Short history: ... by numerical simulations. The true und…
    ...
    Part III: The hard-sphere model
    Short history:
    ...
    by numerical simulations. The true understanding of this transition dates from 2011.
    Molecular Dynamics (animation)
    simulations using the event-driven algorithm.
    Animation

    {Event_chain_box.gif} Molecular Dynamics evolution for four hard disks in a box with walls (simulation by Maxim Berman).
    The Event-driven algorithm (scheme)Scheme
    {Event_movie.jpg} Event-driven Molecular Dynamics simulation for 4 disks in a box
    Here is a cartoon of the time-evolution of four hard disks in a square box. At time t=0, each disk starts at a given position and with a given velocity. The entire time-evolution is simply the solution of Newton's equations: Each disk moves freely until it collides either with another disk or with a wall (see ).
    (view changes)
    4:52 am

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