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76 lines
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ReStructuredText
76 lines
3.0 KiB
ReStructuredText
.. _montecarloloop:
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The Monte Carlo loop
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--------------------
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Introduction
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************
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The ``mc_generic`` class is an implementation of the Monte Carlo loop. Its
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goal is to propose and then accept or reject changes to a configuration
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according to this loop:
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.. image:: loop.png
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:width: 700
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:align: center
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As shown in the figure, after a first initialization, the loop starts by
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proposing an update. In the following, we generically refer to this proposal as
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a proposed **move**. The move proposes a modification of the state of the
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system, which we call the **configuration** of the system. After having
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computed the transition probabilities between this proposed configuration and
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the old one, as well as their probability density, we compute an acceptance
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probability for the move. Based on this probability, the move is either
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**accepted** or **rejected**. If it is rejected, nothing happens and we remain
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in the same configuration. If it is accepted, the configuration is updated.
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This procedure is the heart of the Monte Carlo algorithm and is repeated
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at every Monte Carlo **step** (meaning one loop). Measurements are not
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made at every step, to allow for some decorrelation between measured
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configurations. Thus, measurements are made every :math:`L` steps. We
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say that these :math:`L` steps form a **cycle** and :math:`L` is the
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length of a cycle.
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At the very beginning of the simulation, one usually allows for :math:`W`
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**warmup** (thermalization) cycles. This means that there will be no measurements
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during these first :math:`W` cycles. After that, we define :math:`N`, the
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number of cycles that will be done until the end of the simulation.
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At the end of the simulation, the code will have done:
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* :math:`N` measurements
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* :math:`N + W` cycles
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* :math:`(N + W) \times L` steps
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C++ variable names
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******************
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In the C++ examples, these variables will be called:
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* ``N_Cycles`` :math:`= N`
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* ``Length_Cycle`` :math:`= L`
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* ``N_Warmup_Cycle`` :math:`= W`
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You will also have to use these names if you will construct an ``mc_generic``
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instance from a dictonary (see full reference below).
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Monte Carlo loop and connection with moves and measures
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*******************************************************
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We will cover this in more details, but let us already mention here that the
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``mc_generic`` class only implements the Monte Carlo loop. It doesn't need (and
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in fact doesn't) know anything about what the configuration is or what the
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moves and measurements really do. All it does, is to use external classes which
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take care of making the moves. It just expects back a Metropolis ratio so that
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it can decide wether the move should be accepted or rejected. Once this choice
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is made, it tells the external class which again does we is needed if the move
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is accepted or rejected. The same is true for measurements which are external
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classes called by the loop. This will become clearer with an example in the
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following section.
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.. note::
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Above, we described the Metropolis algorithm. A different accept/reject
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scheme could be used but the mechanism remains the same.
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