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…
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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
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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]
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
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partition function.
* -4-
-4- The condensate
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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
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
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
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partition function.
-4-
* -4- The condensate
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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
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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
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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
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
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with a wall (see ).Inwall. In this simulation,
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on the lefttop is SMAC
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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:
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 forEvent driven dynamics was invented by Alder and Wainwright, in 1957. Here is a cartoon of the time-evolution of four hard
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in a box with walls (simulation by Maxim Berman).
Schemesquare 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,
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wall (see ).
In).In this simulation,
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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.
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 ).