I don't understand this concept very well. Is it like taking the energy each possible configuration and adding them together? For example, if we had a 2D lattice gas and wanted to sum over the "white" the spins in a "checkerboard pattern":
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What are we doing mathematically? Let's say the case of only nearest neighbor interactions and each spin has two states. I understand that each neighbor interaction between an i and j will contribute a term proportional to σiσj to the energy.
But I'm having trouble understanding what we are doing when we "sum over" the spins. Each "white" has 2 states and each of its neighbors have 2 states, so there are 25 different configurations! How do we know which value to take when we sum?
And it only gets more confusing when you consider that those nearest neighbor spins are connected to their neighbors!
Does anyone have a simple, lucid example of "summing over spins" in 2D? I'm studying the RG and this seems related to "block spins" but I'm having trouble with making the connection.
What are we doing mathematically? Let's say the case of only nearest neighbor interactions and each spin has two states. I understand that each neighbor interaction between an i and j will contribute a term proportional to σiσj to the energy.
But I'm having trouble understanding what we are doing when we "sum over" the spins. Each "white" has 2 states and each of its neighbors have 2 states, so there are 25 different configurations! How do we know which value to take when we sum?
And it only gets more confusing when you consider that those nearest neighbor spins are connected to their neighbors!
Does anyone have a simple, lucid example of "summing over spins" in 2D? I'm studying the RG and this seems related to "block spins" but I'm having trouble with making the connection.