basic_tools:phase_space
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basic_tools:phase_space [2018/11/13 22:27] supeng_hong [Intuitive] |
basic_tools:phase_space [2020/05/12 06:25] 173.245.129.4 grammar fix |
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| Try to imagine a | Try to imagine a | ||
| 'space' of a large number of dimensions, one dimension for each of the | 'space' of a large number of dimensions, one dimension for each of the | ||
| - | coordinates $x_1, x_2,\ldots$, $p_1, p_2,\ldots$ (Mathematical spaces often have many; | + | coordinates $x_1, x_2,\ldots$, $p_1, p_2,\ldots$ (Mathematical spaces often have many |
| more than three dimensions.) This space is called phase space (see Fig. 5.10). | more than three dimensions.) This space is called phase space (see Fig. 5.10). | ||
| - | For n unconstrained particles, this be a space of 6n dimensions (three, | + | For n unconstrained particles, this is a space of 6n dimensions (three |
| position coordinates and three momentum coordinates for each particle). The | position coordinates and three momentum coordinates for each particle). The | ||
| reader may well worry that even for a single particle this is already twice as | reader may well worry that even for a single particle this is already twice as | ||
| many dimensions as she or he would normally be used to visualizing! The | many dimensions as she or he would normally be used to visualizing! The | ||
| secret is not to be put off by this. Whereas six dimensions are, indeed, more | secret is not to be put off by this. Whereas six dimensions are, indeed, more | ||
| - | dimensions than can be readily pictured, it would actually not be of much | + | dimensions than can be readily pictured, it would actually not be of much use |
| - | to us if we were in fact able to picture it. For just a room full of | + | to us if we were in fact able to imagine it. For just a room full of |
| molecules, the number of phase-space dimensions might be something like | molecules, the number of phase-space dimensions might be something like | ||
| Line 54: | Line 54: | ||
| is governed by a **flow** in phase space. | is governed by a **flow** in phase space. | ||
| - | To understand this image that you follow one individual trajectory in phase space. Say, you take a pencil, put it down at one point in phase space and then start to draw the correct trajectory for the system which follows from [[equations:hamiltons_equations|Hamilton's equations]]. Now, this trajectory is only one thing that can happen in our system. When we are interested in the bigger picture, we need to follow all trajectories that are possible. This means, that we take an infinite number of pencils, put them down and the draw the trajectories. Each trajectory is calculated with Hamilton's equations. The set of all possible trajectories defines a flow in phase space. | + | To understand this imagine that you follow one individual trajectory in phase space. Say, you take a pencil, put it down at one point in phase space and then start to draw the correct trajectory for the system which follows from [[equations:hamiltons_equations|Hamilton's equations]]. Now, this trajectory is only one thing that can happen in our system. When we are interested in the bigger picture, we need to follow all trajectories that are possible. This means, that we take an infinite number of pencils, put them down and the draw the trajectories. Each trajectory is calculated with Hamilton's equations. The set of all possible trajectories defines a flow in phase space. |
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basic_tools/phase_space.txt · Last modified: 2020/05/12 06:25 by 173.245.129.4
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