basic_notions:boundary_conditions
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basic_notions:boundary_conditions [2017/11/23 13:37] jakobadmin [Student] |
basic_notions:boundary_conditions [2018/03/30 11:13] (current) jakobadmin [Intuitive] |
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| ====== Boundary Conditions ====== | ====== Boundary Conditions ====== | ||
| - | <tabbox Why is it interesting?> | + | <tabbox Intuitive> |
| + | <blockquote>Generically, if we can formulate the [[:equations|equations of motion]] for a theory, we have everything at our disposal to describe the solutions of the theory. However, in general we have to supplement the equations with the situation we want to actually describe with the theory. In the case of the planet, we have to add where the planet was and where it moved to at a certain instance of time. Otherwise the equation of motion would give us the solutions for all possible initial positions and velocities of the planet, and thus an infinite number of possible solutions to the theory. Such additional information are called boundary conditions. They select out of any possible kind of behavior described by a theory the particular one which is compatible with the state a system is in.<cite>http://axelmaas.blogspot.de/2012/03/equations-that-describe-world.html</cite></blockquote> | ||
| + | |||
| + | <tabbox Concrete> | ||
| + | **Important Spatial Boundary Conditions:** | ||
| + | -->Overview# | ||
| + | <blockquote>To specify the lattice model we must prescribe the boundary | ||
| + | conditions for the scalar field. These conditions are classified as follows: | ||
| - | <blockquote> | + | * Periodic boundary conditions: With these conditions the lattice is a discrete torus and the lattice field theory is invariant under discrete translations and rotations. |
| - | The [[:equations|field equations]] and the boundary | + | * Fixed boundary conditions: Here we prescribe the field on the boundary $φ|_{∂Λ}$. Such boundary conditions are useful to describe entangled states in quantum field theory. |
| - | conditions are inextricably connected and the latter can in no way be considered less important | + | * Open boundary conditions: Here we switch off all interactions between sites on the lattice Λ with sites in the complement of Λ (viewed as subset of $\mathbb Z_d$ ). These boundary conditions are used in solid state physics. |
| - | than the former<cite>V. Fock, The theory of space, time and gravitation</cite></blockquote> | + | * Antiperiodic boundary conditions: They serve as a tool to inhibit unwanted longrange correlations or to study interfaces. This modification of the periodic boundary conditions is frequently used in lattice field theories. |
| - | + | <cite>Statistical Approach to Quantum Field Theory by Andreas Wipf</cite> | |
| - | <blockquote>Now, Nature is described by fields, and this elegant and powerful formulation of classical | + | |
| - | and quantum mechanics based on the action needs to be supplemented with a careful treatment | + | |
| - | of boundary conditions at infinity. The issue of boundary conditions is particularly important | + | |
| - | and interesting in the case of [[advanced_tools:gauge_symmetry|gauge theories]] where the assumption ‘all fields decay sufficiently | + | |
| - | rapidly at infinity’ is not justified. <cite>https://arxiv.org/pdf/1601.03616.pdf</cite> | + | |
| </blockquote> | </blockquote> | ||
| + | <-- | ||
| - | <blockquote>[I]t is natural to regulate infinite sized systems by imposing boundary | + | -->Periodic Boundary Conditions# |
| - | conditions at finite distance, often described as placing the system in a box. This idea | + | |
| - | has a long history in the gravitational context (see e.g. [15–27]) where it is common to | + | |
| - | impose a Dirichlet boundary condition, fixing the induced metric at the walls of the box1 | + | |
| - | .<cite>https://arxiv.org/abs/1508.02515</cite></blockquote> | + | |
| + | <note tip>Eliminates surfaces and is the most popular choice of boundary conditions | ||
| + | </note> | ||
| + | <-- | ||
| + | |||
| + | |||
| + | **Important Boundary Conditions for Differential Equations:** | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
| + | |||
| + | --> Dirichlet boundary condition# | ||
| + | <note tip>Dirichlet = Data on boundary</note> | ||
| + | |||
| + | - The value of the dependent variable is specified on the boundary. | ||
| + | |||
| + | - Needed for elliptic or parabolic partial differential equations. Other boundary conditions are insufficient to determine a unique solution, overly restrictive, or lead to instabilities. | ||
| + | |||
| + | - "In thermodynamics, Dirichlet boundary conditions consist of surfaces (in 3D problems) held at a fixed temperatures." ([[https://www.simscale.com/docs/content/simwiki/numerics/what-are-boundary-conditions.html|Source]]) | ||
| + | |||
| + | <-- | ||
| + | |||
| + | -->Neumann boundary condition# | ||
| + | |||
| + | <note tip>Neumann = Normal derivative on boundary</note> | ||
| + | |||
| + | - The normal derivative of the dependent variable is specified on the boundary. | ||
| + | |||
| + | - Needed for elliptic or parabolic partial differential equations. Other boundary conditions are insufficient to determine a unique solution, overly restrictive, or lead to instabilities. | ||
| + | |||
| + | - "In thermodynamics, the Neumann boundary condition represents the heat flux across the boundaries." ([[https://www.simscale.com/docs/content/simwiki/numerics/what-are-boundary-conditions.html|Source]]) | ||
| + | |||
| + | <-- | ||
| + | |||
| + | -->Cauchy boundary condition# | ||
| + | |||
| + | <note tip>Cauchy = Dirichlet $\oplus$ Neumann</note> | ||
| + | |||
| + | - Both the value and the normal derivative of the dependent variable are specified on the boundary. | ||
| + | |||
| + | |||
| + | |||
| + | - Cauchy boundary conditions are analogous to the initial conditions for a second-order ordinary differential equation. | ||
| + | |||
| + | - Needed for Hyperbolic equations on an open surface. Other boundary conditions are either too restrictive for a solution to exist, or insufficient to determine a unique solution. | ||
| + | |||
| + | - In physics needed for classical and quantum field theory. | ||
| + | <-- | ||
| + | |||
| + | -->Robin boundary condition# | ||
| + | |||
| + | <note tip>Robin = only a condition on linear combination of Dirichlet and Neumann. Not Dirichlet + Neumann!</note> | ||
| + | |||
| + | - The the value of a linear combination of the dependent variable and the normal derivative of the dependent variable is specified on the boundary. | ||
| + | |||
| + | <-- | ||
| - | <tabbox Layman> | ||
| - | <note tip> | ||
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | ||
| - | </note> | ||
| - | | ||
| - | <tabbox Student> | ||
| - | **Important Boundary Conditions:** | ||
| - | * Dirichlet boundary condition | ||
| - | * The value of the dependent variable is specified on the boundary. | ||
| - | * Needed for elliptic or parabolic partial differential equations. Other boundary conditions are insufficient to determine a unique solution, overly restrictive, or lead to instabilities. | ||
| - | * "In thermodynamics, Dirichlet boundary conditions consist of surfaces (in 3D problems) held at a fixed temperatures." ([[https://www.simscale.com/docs/content/simwiki/numerics/what-are-boundary-conditions.html|Source]]) | ||
| - | * Neumann boundary condition | ||
| - | * The normal derivative of the dependent variable is specified on the boundary. | ||
| - | * Needed for elliptic or parabolic partial differential equations. Other boundary conditions are insufficient to determine a unique solution, overly restrictive, or lead to instabilities. | ||
| - | * "In thermodynamics, the Neumann boundary condition represents the heat flux across the boundaries." ([[https://www.simscale.com/docs/content/simwiki/numerics/what-are-boundary-conditions.html|Source]]) | ||
| - | * Cauchy boundary condition | ||
| - | * Both the value and the normal derivative of the dependent variable are specified on the boundary. | ||
| - | * Schematically: Cauchy = Dirichlet $\oplus$ Neumann. | ||
| - | * Cauchy boundary conditions are analogous to the initial conditions for a second-order ordinary differential equation. | ||
| - | * Needed for Hyperbolic equations on an open surface. Other boundary conditions are either too restrictive for a solution to exist, or insufficient to determine a unique solution. | ||
| - | * In physics needed for classical and quantum field theory. | ||
| - | * Robin boundary condition. | ||
| - | * The the value of a linear combination of the dependent variable and the normal derivative of the dependent variable is specified on the boundary. | ||
| - | * Robin = only linear combination of Dirichlet and Neumann. Not Dirichlet $\oplus$ Neumann! | ||
| - | * Mixed boundary condition | ||
| **Recommended Resources:** | **Recommended Resources:** | ||
| Line 61: | Line 93: | ||
| * For boundary conditions in gauge field theory, see section 4.5 "Cauchy problem and gauge conditions" in Rubakov's "Classical Theory of Gauge Fields". | * For boundary conditions in gauge field theory, see section 4.5 "Cauchy problem and gauge conditions" in Rubakov's "Classical Theory of Gauge Fields". | ||
| - | <tabbox Researcher> | + | <tabbox Abstract> |
| <note tip> | <note tip> | ||
| Line 68: | Line 100: | ||
| | | ||
| - | <tabbox Examples> | + | <tabbox Why is it interesting?> |
| - | --> Example1# | ||
| - | |||
| - | <-- | ||
| - | --> Example2:# | + | <blockquote> |
| + | The [[:equations|field equations]] and the boundary | ||
| + | conditions are inextricably connected and the latter can in no way be considered less important | ||
| + | than the former<cite>V. Fock, The theory of space, time and gravitation</cite></blockquote> | ||
| - | + | <blockquote>Now, Nature is described by fields, and this elegant and powerful formulation of classical | |
| - | <-- | + | and quantum mechanics based on the action needs to be supplemented with a careful treatment |
| - | + | of boundary conditions at infinity. The issue of boundary conditions is particularly important | |
| - | <tabbox History> | + | and interesting in the case of [[advanced_tools:gauge_symmetry|gauge theories]] where the assumption ‘all fields decay sufficiently |
| + | rapidly at infinity’ is not justified. <cite>https://arxiv.org/pdf/1601.03616.pdf</cite> | ||
| + | </blockquote> | ||
| + | |||
| + | <blockquote>[I]t is natural to regulate infinite sized systems by imposing boundary | ||
| + | conditions at finite distance, often described as placing the system in a box. This idea | ||
| + | has a long history in the gravitational context (see e.g. [15–27]) where it is common to | ||
| + | impose a Dirichlet boundary condition, fixing the induced metric at the walls of the box1 | ||
| + | .<cite>https://arxiv.org/abs/1508.02515</cite></blockquote> | ||
| </tabbox> | </tabbox> | ||
basic_notions/boundary_conditions.1511440658.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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