Summing Out States: Where Did the i_1 Sum Go?

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In summary, when discussing the concept of "summing out" the states of all other particles, it is important to note that the state of one particular particle, i_1, is kept unspecified while the states of all other particles are summed over. This can be compared to an analogy of drawing balls from vases, where the first vase is fixed and the remaining vases are summed over to determine the fraction of configurations with a particular state. This concept is important in statistical mechanics when considering the Boltzmann factor.
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latentcorpse
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At the top of page 26 here
http://www.ph.ed.ac.uk/~pmonthou/Statistical-Mechanics/documents/SM7.pdf

when we talk about "summing out" the states of all the other particles, why are we not summing over [itex]i_1[/itex] in the following sum - where did the [itex]i_1[/itex] sum go to?
 
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The idea is that you are keeping [itex]i_1[/itex] unspecified, but you are summing over the state of all the other particles.

Maybe the following analogy will help you. Suppose that I have 10 vases with blue and red balls, and I draw a ball from each of them. In principle, I have to specify a "state" of the "system" by saying for each vase, whether I draw a red or a blue ball, for example: (BBRBRRBRRB). Now suppose that I am interested in the first one only, and I want to calculate how many systems there are in which I have a red or a blue ball in vase 1.
Let Z be the total number of possible configurations (in this case, clearly 2^10). To find out how many arrangements there are in which the first ball drawn is blue, I can simply sum over all configurations of numbers 2 -- 10, while the first one is fixed. So I get
(BRRRRRRRRRR), (BRRRRRRRRB), (BRRRRRRRBR), (BRRBRBRRBRB), etc.
and I need to count all of them.
So what I will get is
[tex]N = \sum_{i_2 = B, R} \sum_{i_3 = B, R} \cdots \sum_{i_{10} = B, R} 1[/tex]
and the fraction (relative amount) of configurations with a blue one in the first vase is N / Z (which in this case, of course, will simply give 1/2).

The example you posted is very similar, only there every configuration isn't equivalent, but you have to weigh it by some Boltzmann factor.
 

FAQ: Summing Out States: Where Did the i_1 Sum Go?

What is "Summing Out States" and why is it important in scientific research?

Summing Out States is a mathematical technique used in statistical mechanics and physics to simplify complex calculations. It allows scientists to reduce the number of variables in a system, making it easier to analyze and understand. This technique is especially important in studying systems with a large number of particles or variables, as it helps to reduce computational complexity.

How does "Summing Out States" work?

In "Summing Out States", the goal is to eliminate a variable or set of variables from a system by integrating over all possible values of those variables. This is done by using the concept of marginalization, where the probability distribution of the remaining variables is calculated by summing over the eliminated variables. This technique is based on the principle of conservation of probability, which states that the total probability of all possible outcomes in a system must equal 1.

What are the benefits of using "Summing Out States" in scientific research?

The main benefit of "Summing Out States" is that it simplifies complex calculations, making it easier to analyze and understand a system. This technique also helps to reduce computational complexity, which can be especially useful when dealing with large datasets or systems with a large number of variables. Additionally, "Summing Out States" can help to identify and highlight key variables that have the most significant impact on the overall system.

What are the limitations of "Summing Out States"?

One of the main limitations of "Summing Out States" is that it may not always be possible to eliminate all variables from a system. In some cases, certain variables may be interdependent and cannot be eliminated individually. Additionally, this technique may not be suitable for systems with non-linear relationships between variables or for situations where the number of variables is very large.

How is "Summing Out States" used in different fields of science?

"Summing Out States" is a widely used technique in various fields of science, including statistical mechanics, physics, computer science, and machine learning. In statistical mechanics, it is used to calculate the partition function of a system, while in physics, it is used in thermodynamics and quantum mechanics. In computer science and machine learning, it is used to simplify calculations and improve the efficiency of algorithms. Additionally, "Summing Out States" has applications in fields such as biology, economics, and social sciences.

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