✎ Boundary Conditions [The Physics Travel Guide]

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====== Boundary Conditions ======

<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 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:

* Periodic boundary conditions: With these conditions the lattice is a discrete torus and the lattice field theory is invariant under discrete translations and rotations. 
  * 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. 
  * 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. 
  * 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>
<--

-->Periodic Boundary Conditions#

<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.

<--

**Recommended Resources:**

* https://www.simscale.com/docs/content/simwiki/numerics/what-are-boundary-conditions.html
  * For boundary conditions for waves, see chapter 9 in Georgi's "[[http://www.people.fas.harvard.edu/~hgeorgi/onenew.pdf|THE PHYSICS OF WAVES]]"
  * 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 Abstract>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

<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
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>

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