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Universiteit Utrecht

1. Wijn, Astrid Silvia de. Chaos in systems with many degrees of freedom.

Degree: 2004, Universiteit Utrecht

URL: http://dspace.library.uu.nl:8080/handle/1874/1162

In this thesis I discuss some of the chaotic properties specific to systems
of many particles and other systems with many degrees of freedom.
A dynamical system is called chaotic if a typical infinite perturbation of
initial conditions grows exponentially with time.
The chaoticity of a system is characterised by the Lyapunov exponents, which
indicate the possible rates at which an infinitesimal perturbation of
initial conditions may grow or decrease.
A system has as many Lyapunov exponents as it's phase space has dimensions,
and so a system with many degrees of freedom has many Lyapunov exponents.
A system is chaotic if it has at least one positive exponent.
The sum of the positive Lyapunov exponents equals the maximal rate of
information increase of the system and is referred to as the
Kolmogorov-Sinai entropy.
The dynamical properties of a system are thought to have bearing on the
non-equilibrium behaviour.
This thesis contains calculations of Lyapunov exponents of three different
systems; namely, systems consisting of many, freely moving hard disks and
hard spheres are discussed,
as well as the Lorentz gas, which is a system consisting of fixed spherical
scatterers with a point particle moving between and colliding elastically
with them, and a similar system in which the scatterers are cylindrical.
Chapter 3 contains calculations of the smallest positive and negative
exponents of hard disks.
These are known from simulations to have interesting behaviour, if the
system is large enough.
They are explained as belonging to Goldstone modes associated with the
symmetries of the tangent space, and the Lyapunov exponents are calculated.
This allows their calculation by formulating a generalised Boltzmann
equation and solving this perturbatively.
There is a slight discrepancy between the predicted values and simulation
results, which may be attributed to the neglect of ring-collision terms in
the Boltzmann equation.
In chapter 4 the Kolmogorov-Sinai entropy is calculated for the same system.
It is known to be proportional to the collision frequency multiplied by a
factor that equals the logarithm of the density plus a constant.
Previous calculations of this constant have produced unsatisfactory results.
I discuss the cause of this and how to calculate the constant correctly.
A system of freely moving hard particles can be described as a point
particle in a high-dimensional space, colliding with cylindrical scatterers
with very specific positions and orientations.
Because this is very similar to the Lorentz gas, where the scatterers are
hyperspheres, in chapter 5 I examine the Lyapunov exponents of the Lorentz
gas in an arbitrary number of dimensions.
The full spectrum is calculated analytically and the similarities and
differences between the high-dimensional Lorentz gas and systems of hard
disks or spheres are discussed.
In chapter 6 I describe numerical calculations of the Lyapunov exponents of
systems consisting of isotropically oriented, homogeneously…

Subjects/Keywords: Natuur- en Sterrenkunde; chaos; hard disks; hard spheres; Lorentz gas; dynamical system; statistical mechanics; kinetic theory; Lyapunov exponent; Kolmogorov-Sinai entropy; Goldstone mode; Boltzmann equation

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Wijn, A. S. d. (2004). Chaos in systems with many degrees of freedom. (Doctoral Dissertation). Universiteit Utrecht. Retrieved from http://dspace.library.uu.nl:8080/handle/1874/1162

Chicago Manual of Style (16^{th} Edition):

Wijn, Astrid Silvia de. “Chaos in systems with many degrees of freedom.” 2004. Doctoral Dissertation, Universiteit Utrecht. Accessed January 20, 2020. http://dspace.library.uu.nl:8080/handle/1874/1162.

MLA Handbook (7^{th} Edition):

Wijn, Astrid Silvia de. “Chaos in systems with many degrees of freedom.” 2004. Web. 20 Jan 2020.

Vancouver:

Wijn ASd. Chaos in systems with many degrees of freedom. [Internet] [Doctoral dissertation]. Universiteit Utrecht; 2004. [cited 2020 Jan 20]. Available from: http://dspace.library.uu.nl:8080/handle/1874/1162.

Council of Science Editors:

Wijn ASd. Chaos in systems with many degrees of freedom. [Doctoral Dissertation]. Universiteit Utrecht; 2004. Available from: http://dspace.library.uu.nl:8080/handle/1874/1162

2. Wijn, Astrid Silvia de. Chaos in systems with many degrees of freedom.

Degree: 2004, University Utrecht

URL: http://dspace.library.uu.nl/handle/1874/1162 ; URN:NBN:NL:UI:10-1874-1162 ; urn:isbn:90-393-3868-X ; URN:NBN:NL:UI:10-1874-1162 ; http://dspace.library.uu.nl/handle/1874/1162

In this thesis I discuss some of the chaotic properties specific to systems
of many particles and other systems with many degrees of freedom.
A dynamical system is called chaotic if a typical infinite perturbation of
initial conditions grows exponentially with time.
The chaoticity of a system is characterised by the Lyapunov exponents, which
indicate the possible rates at which an infinitesimal perturbation of
initial conditions may grow or decrease.
A system has as many Lyapunov exponents as it's phase space has dimensions,
and so a system with many degrees of freedom has many Lyapunov exponents.
A system is chaotic if it has at least one positive exponent.
The sum of the positive Lyapunov exponents equals the maximal rate of
information increase of the system and is referred to as the
Kolmogorov-Sinai entropy.
The dynamical properties of a system are thought to have bearing on the
non-equilibrium behaviour.
This thesis contains calculations of Lyapunov exponents of three different
systems; namely, systems consisting of many, freely moving hard disks and
hard spheres are discussed,
as well as the Lorentz gas, which is a system consisting of fixed spherical
scatterers with a point particle moving between and colliding elastically
with them, and a similar system in which the scatterers are cylindrical.
Chapter 3 contains calculations of the smallest positive and negative
exponents of hard disks.
These are known from simulations to have interesting behaviour, if the
system is large enough.
They are explained as belonging to Goldstone modes associated with the
symmetries of the tangent space, and the Lyapunov exponents are calculated.
This allows their calculation by formulating a generalised Boltzmann
equation and solving this perturbatively.
There is a slight discrepancy between the predicted values and simulation
results, which may be attributed to the neglect of ring-collision terms in
the Boltzmann equation.
In chapter 4 the Kolmogorov-Sinai entropy is calculated for the same system.
It is known to be proportional to the collision frequency multiplied by a
factor that equals the logarithm of the density plus a constant.
Previous calculations of this constant have produced unsatisfactory results.
I discuss the cause of this and how to calculate the constant correctly.
A system of freely moving hard particles can be described as a point
particle in a high-dimensional space, colliding with cylindrical scatterers
with very specific positions and orientations.
Because this is very similar to the Lorentz gas, where the scatterers are
hyperspheres, in chapter 5 I examine the Lyapunov exponents of the Lorentz
gas in an arbitrary number of dimensions.
The full spectrum is calculated analytically and the similarities and
differences between the high-dimensional Lorentz gas and systems of hard
disks or spheres are discussed.
In chapter 6 I describe numerical calculations of the Lyapunov exponents of
systems consisting of isotropically oriented, homogeneously…

Subjects/Keywords: chaos; hard disks; hard spheres; Lorentz gas; dynamical system; statistical mechanics; kinetic theory; Lyapunov exponent; Kolmogorov-Sinai entropy; Goldstone mode; Boltzmann equation

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Wijn, A. S. d. (2004). Chaos in systems with many degrees of freedom. (Doctoral Dissertation). University Utrecht. Retrieved from http://dspace.library.uu.nl/handle/1874/1162 ; URN:NBN:NL:UI:10-1874-1162 ; urn:isbn:90-393-3868-X ; URN:NBN:NL:UI:10-1874-1162 ; http://dspace.library.uu.nl/handle/1874/1162

Chicago Manual of Style (16^{th} Edition):

Wijn, Astrid Silvia de. “Chaos in systems with many degrees of freedom.” 2004. Doctoral Dissertation, University Utrecht. Accessed January 20, 2020. http://dspace.library.uu.nl/handle/1874/1162 ; URN:NBN:NL:UI:10-1874-1162 ; urn:isbn:90-393-3868-X ; URN:NBN:NL:UI:10-1874-1162 ; http://dspace.library.uu.nl/handle/1874/1162.

MLA Handbook (7^{th} Edition):

Wijn, Astrid Silvia de. “Chaos in systems with many degrees of freedom.” 2004. Web. 20 Jan 2020.

Vancouver:

Wijn ASd. Chaos in systems with many degrees of freedom. [Internet] [Doctoral dissertation]. University Utrecht; 2004. [cited 2020 Jan 20]. Available from: http://dspace.library.uu.nl/handle/1874/1162 ; URN:NBN:NL:UI:10-1874-1162 ; urn:isbn:90-393-3868-X ; URN:NBN:NL:UI:10-1874-1162 ; http://dspace.library.uu.nl/handle/1874/1162.

Council of Science Editors:

Wijn ASd. Chaos in systems with many degrees of freedom. [Doctoral Dissertation]. University Utrecht; 2004. Available from: http://dspace.library.uu.nl/handle/1874/1162 ; URN:NBN:NL:UI:10-1874-1162 ; urn:isbn:90-393-3868-X ; URN:NBN:NL:UI:10-1874-1162 ; http://dspace.library.uu.nl/handle/1874/1162

3. Yu, Yi-Xiang. Superfluids of Fermions in spin-orbit coupled systems and photons inside a cavity.

Degree: PhD, Physics and Astronomy, 2015, Mississippi State University

URL: http://sun.library.msstate.edu/ETD-db/theses/available/etd-05142015-120012/ ;

This dissertation introduces some new properties of both superfluid phases of fermions
with spin-orbit coupling (SOC) and superradiant phases of photons in an optical cavity. The
effects of SOC on the phase transition between normal and superfluid phase are revealed;
an unconventional crossover driven by SOC from the Bardeen-Cooper-Schrieffer (BCS)
state to the Bose-Einstein condensate (BEC) state is verified in three different systems; and
two kinds of excitations, a Goldstone mode and a Higgs mode, are demonstrated to occur
in a quantum optical system.
We investigate the BCS superfluid state of two-component atomic Fermi gases in the
presence of three kinds of SOCs. We find that SOC drives a class of BCS to BEC crossover
that is different from the conventional one without SOC. Here, we extend the concepts of
the coherence length and Cooper-pair size in the absence of SOC to Fermi systems with
SOC. We study the dependence of chemical potential, coherence length, and Cooper-pair
size on the SOC strength and the scattering length in three dimensions (3D) (or the twobody
binding energy in two dimensions (2D)) for three attractively interacting Fermi gases
with 3D Rashba, 3D Weyl, and 2D Rashba SOC respectively.
By adding a population imbalance to a Fermi gas with Rashba-type SOC, we also map
out the finite-temperature phase diagram. Due to a competition between SOC and population
imbalance, the finite-temperature phase diagram reveals a large variety of new features,
including the expanding of the superfluid state regime and the shrinking of both the phase
separation and the normal regimes. We find that the tricritical point moves toward a regime
of low temperature, high magnetic field, and high polarization as the SOC strength increases.
Besides Fermi fluids, this dissertation also gives a new angle of view on the superradiant
phase in the Dicke model. Here, we demonstrate that Goldstone and Higgs modes can be
observed in an optical system with only a few atoms inside a cavity. The model we study
is the <i>U</i>(1)/<i>Z</i>_{2} Dicke model with N qubits (two-level atoms) coupled to a single photon
mode.
*Advisors/Committee Members: Dr. Yaroslav Koshka (committee member), Dr. R. Torsten Clay (committee member), Dr. Steven R. Gwaltney (committee member), Dr. Seong-Gon Kim (committee member), Dr. Jinwu Ye (chair).*

Subjects/Keywords: Cooper-pair size; superradiant phase; Dicke model; Goldstone mode; Higgs mode; phase diagram; Bose-Einstein condensate; Fermi gas; spin-orbit coupling

…the Higgs *mode* and pseudo-*Goldstone* *mode* in
one-gap and two-gap superconductors… …62
4.3
The analytical *Goldstone* *mode* at α = −1/2, EG (α = −1/2) = D(g)… …The analytical spectral weight (red) of the *Goldstone* *mode* CG against the
ED… …45
47
48
49
54
4. *GOLDSTONE* AND HIGGS MODES OF PHOTONS . . . . . . . . . . . 57
4.1
4.2… …Reducing the U (1)/Z2 to the J − U (1)/Z2 Dicke model . . . . .
*Goldstone* and…

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Yu, Y. (2015). Superfluids of Fermions in spin-orbit coupled systems and photons inside a cavity. (Doctoral Dissertation). Mississippi State University. Retrieved from http://sun.library.msstate.edu/ETD-db/theses/available/etd-05142015-120012/ ;

Chicago Manual of Style (16^{th} Edition):

Yu, Yi-Xiang. “Superfluids of Fermions in spin-orbit coupled systems and photons inside a cavity.” 2015. Doctoral Dissertation, Mississippi State University. Accessed January 20, 2020. http://sun.library.msstate.edu/ETD-db/theses/available/etd-05142015-120012/ ;.

MLA Handbook (7^{th} Edition):

Yu, Yi-Xiang. “Superfluids of Fermions in spin-orbit coupled systems and photons inside a cavity.” 2015. Web. 20 Jan 2020.

Vancouver:

Yu Y. Superfluids of Fermions in spin-orbit coupled systems and photons inside a cavity. [Internet] [Doctoral dissertation]. Mississippi State University; 2015. [cited 2020 Jan 20]. Available from: http://sun.library.msstate.edu/ETD-db/theses/available/etd-05142015-120012/ ;.

Council of Science Editors:

Yu Y. Superfluids of Fermions in spin-orbit coupled systems and photons inside a cavity. [Doctoral Dissertation]. Mississippi State University; 2015. Available from: http://sun.library.msstate.edu/ETD-db/theses/available/etd-05142015-120012/ ;