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This question uses some of the terminology of quantum mechanics, but it is fundamentally a maths question. If you recognize the physics, I am trying to justify a system tending towards the same unique equilibrium regardless of the initial state.
Let finite sequences of 0s and 1s form the basis of a vector space. For example, is a element of this basis.
Let be a unitary operator on this vector space. This operator essentially "appends a 0 or 1", but to a linear combination. One example is:
For any sequence s. Think of U as the next-time-step operator.
Let pick all the coefficients of basis vectors that form v that end with x, and takes the sum of their norm-squared. For example,
and .
Now, consider for some initial v. It can be shown that for any initial v and either x.
Furthermore, if you consider sequences of some finite element space with m elements (beyond just 2 elements as in this example) then . So the equilibrium distribution is uniform over the element space, and is independent of the initial state.
What I am interested in is when the element space becomes infinite. I suspect that the equilibrium distribution is not uniform over the element space anymore in that case. Is the quantity still independent of the initial v?
-- 49.183.55.236 ( talk) 05:30, 8 May 2019 (UTC)
Mathematics desk | ||
---|---|---|
< May 7 | << Apr | May | Jun >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
This question uses some of the terminology of quantum mechanics, but it is fundamentally a maths question. If you recognize the physics, I am trying to justify a system tending towards the same unique equilibrium regardless of the initial state.
Let finite sequences of 0s and 1s form the basis of a vector space. For example, is a element of this basis.
Let be a unitary operator on this vector space. This operator essentially "appends a 0 or 1", but to a linear combination. One example is:
For any sequence s. Think of U as the next-time-step operator.
Let pick all the coefficients of basis vectors that form v that end with x, and takes the sum of their norm-squared. For example,
and .
Now, consider for some initial v. It can be shown that for any initial v and either x.
Furthermore, if you consider sequences of some finite element space with m elements (beyond just 2 elements as in this example) then . So the equilibrium distribution is uniform over the element space, and is independent of the initial state.
What I am interested in is when the element space becomes infinite. I suspect that the equilibrium distribution is not uniform over the element space anymore in that case. Is the quantity still independent of the initial v?
-- 49.183.55.236 ( talk) 05:30, 8 May 2019 (UTC)