Stochastic Process with Memory

Full Title

A Stochastic Process with Memory refers to a random process in which the future state depends not only on the present state but also on past states. This contrasts with memoryless (Markov) processes, where the future depends solely on the current state.

The notion of long range dependence has, clearly, something to do with memory in a stochastic process. "The 'specialness' of long memory indicates that most stationary stochastic processes do not have it. it appears that connecting the notion of long range dependence to certain types of phase transitions is promising. It fits well with intuition of the term “long memory” describing a model that is out of the ordinary. Furthermore, it allows us to concentrate on the behavior of really important functionals. Much remains to be done to clarify both possible types of such Phase Transitions and the relevant boundaries for concrete families of stochastic processes."

A stationary process is a stochastic process whose unconditional joint probability distribution does not change when shifted in time.

Clearly a photon that has originated in the early universe has a very long memory by now.

Context

The first serious argument that this can be important is in a series of papers of B. Mandelbrot and his co-authors, e.g. Mandelbrot and Mandelbrot and Wallis. It is also due to the influence of these early papers and subsequent publications of Mandelbrot (especially Mandelbrot) that long range dependence has also become associated with scaling and fractal behavior.

Is physics markovian

https://www.nature.com/articles/nphys2085

Quantum Physics

In quantum physics, "long range dependencies" refers to situations where quantum particles, even when separated by large distances, can still be interconnected and influence each other's behavior due to a phenomenon called entanglement, meaning their quantum states are linked, or Entangled, leading to non-local correlations that defy classical physics interpretations; this is particularly relevant in systems with long-range interactions, like certain types of atomic arrays or systems with dipole moments, where particles can interact significantly over large distances. Key points about long range dependencies in quantum physics:

• Entanglement: The primary mechanism behind long-range dependencies is entanglement, where two or more particles are linked in a way that measuring the state of one instantly affects the state of the others, regardless of the distance separating them.
• Examples of systems with long-range interactions:
  • Rydberg atoms: Highly excited atoms with large dipole moments that can interact strongly over long distances.
  • Trapped ion systems: Ions held in electromagnetic traps can exhibit long-range interactions through the Coulomb force.
  • Dipolar systems: Systems where particles have dipole moments, like magnetic dipoles, can interact over larger distances.

Impact on quantum phenomena: Long-range interactions can lead to unique quantum phenomena like:

• Non-classical correlations: Violations of Bell inequalities, demonstrating correlations beyond what classical physics allows.
• Exotic quantum phases: Emergence of novel quantum phases not seen in systems with only short-range interactions.
• Quantum criticality: Sharp changes in system behavior at specific points in parameter space due to long-range interactions.

Applications of long-range dependencies:

• Quantum computing: Entanglement is a key resource for quantum computers, allowing for powerful quantum algorithms that can solve certain problems much faster than classical computers.
• Quantum simulation: Studying complex quantum systems with long-range interactions can be achieved using controlled quantum systems in labs to understand phenomena like superconductivity or magnetism.
• Quantum communication: Entanglement can be used to secure quantum communication channels by exploiting the non-local correlations.
• Long-range interacting quantum systems