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title: Time averages vs ensemble averages
tags: [Templates, ' Talk']

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title: Time averages vs ensemble averages
tags: Templates, Talk
description: MD meeting time averages vs ensemble averages.
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# Time averages vs ensemble averages
Christian Blau (blau@kth.se)

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## How to split 1µs of simulation time:

- 1 x 1µs
- 2 x 500ns
- ...
- 256 x 3.90625 ns

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## Pros for each


| 1 x 1µs  | <=> | 256 x 3.90625 ns |
| -------- |---|-------- | -------- |
| simple setup |  |easy error estimates |
| long wait to result | | easy to find compute slots |
| exploration long from start | | guarantee to stay close to start |
| capture slow processes | | know how representable trajectories are |

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## Some limitations 

The best choice is limited by
- what you want to observe
- how accuratly slow processes may be estimated from short simulations
- where you start from
- how you perform your simulation - pulling, pushing etc

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## Slow processes

- ligand unbinding 
- opening-closing transitions
- FRET energy transfer simulation

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## Inadvertedly non-equilibium

- always when starting from a single starting point
- pulling
- waiting 

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## Paper to address these situations very generally

- Alessio Lapolla, David Hartich, and Aljaž Godec: [Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.043084#fulltext)

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## Berg-Purcell, single chemoreceptor
- What are the physical limits for cell to determine ligand concentrations? see 1977 [Physics of chemoreception](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1473391/)


![](https://i.imgur.com/2TMNpgA.png)

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## Single cell receptor - find c!

![](https://i.imgur.com/FuaGa0U.png)
 
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## Everything is in local time fraction.?

- All information about a system 

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## Extremely short measuring time 

- few receptors: (0,1), (1,0), (1,0)
    -  binomial statistics
-  more receptors: (0,1), (1,0), ..., (1,0)
    -  Poisson statistics
-  many receptors $N$:
    -  Gaussian statistic
    -  $\sigma^2\propto1/N$ (Central limit theorem - combinations of measurements become Gaussian)

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## Long measuring time

- $\sigma\propto\frac{1}{t}$ (Central limit theorem), think of splitting trajectory into independent parts

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## In-between measuring time

![](https://i.imgur.com/dQm09aa.png)

let's have a look at the [paper](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.043084#fulltext)!

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