我有这个main.tex文件，编译时没有问题。突然 TeXstudio 崩溃了，现在相同的代码出现了很多错误，我真的不明白为什么。
请问有人能告诉我问题出在哪里吗？也许我删除了一些软件包或其他东西？
此外，它还通过在崩溃前没有的所有地方放置空格来改变整个代码的布局。
如果有人能帮助我避免失去迄今为止所做的一切，我将不胜感激。
\documentclass[a4paper,12pt]{book}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{hyperref}
\usepackage{fancyhdr}
\usepackage{bm} % {\bm ... }
\usepackage[absolute,overlay]{textpos} % textblock
\usepackage{dsfont} %Identity operator
\usepackage{tikz}
\usepackage{pgfplots}
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\usepackage{fancyhdr}
\pagestyle{fancy}
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\fancyhead[LE,RO]{\thepage}
\fancyhead[LO]{\itshape{\rightmark}}
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%%%%%%%%%%%%%%%% Environment for dedication Page %%%%%%%%%%%%%%%%%%
\newenvironment{dedication}
{\clearpage % we want a new page
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\vspace*{\stretch{0.5}}% some space at the top
\itshape % the text is in italic
\raggedleft % flush to the right margin
}
{\par % end the paragraph
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% senza asterisco sezione numerata con asterisco non numerata
% ATTENZIONE: se non è numerata non compare in table of contents!
\begin{document}
\frontmatter % greek numbering
\vspace*{\fill}
\section*{\centering Abstract}
\hspace{\parindent}
Renormalization group analysis is a useful tool for studying critical behaviour of stochastic systems. In this thesis, field theoretic renormalization group and the operator product expansion will be applied to the scalar model representing directed percolation, known as Gribov model, in presence of the random velocity field. Turbulent mixing will be modelled by the compressible form of stochastic Navier-Stokes equation. The task will be to find corresponding critical exponents and scaling functions.
\vspace*{\fill}
%\section*{\centering Svensk sammanfattning}
\thispagestyle{empty}
\newpage
\vspace*{\fill}
\section*{\centering Acknowledgements}
\hspace{\parindent}
{\centering First of all I would like to thank my supervisors: Ralf Eichhorn, in Stockholm, and Miguel Onorato, in Turin, without whom this would not be possible.
Moreover, I would like to thank NORDITA, the Nordic Institute for Theoretical Physics, which hosted me for 6 months and allowed me to work properly on my thesis in a wonderful environment and to get in touch with a lot of interesting people.
In addition to that, I want to thank some people from Nordita whom gave me a lot of useful advices and helped me to manage the work in the first beginning: Stefano Bo (Nordita), Raffaele Marino (Nordita), Lorenzo Manganaro (Unito) and Viktor Skultety (Nordita).\linebreak
Lastly, I would love to thank my family, my mother and my grandparents since they have financially supported these 6 months in Sweden and encouraged me everyday in pursuing my goals through any kind of difficulties. }
\vspace*{\fill}
\thispagestyle{empty}
\begin{dedication}
To my family:\\
My mother, grandfather and grandmother.
\end{dedication}
\tableofcontents
\newpage
\listoftables
\newpage
\listoffigures
%%%%%%%%%%%%%% Introduction Starts %%%%%%%%%%%%%%%%%%%%%
\chapter*{Introduction}
\addcontentsline{toc}{chapter}{Introduction}
\section*{The Brownian Motion }
The birth of the theory of stochastic processes can be clearly related to the study of a well known physical phenomenon which has been studied among centuries and still nowadays leads to interesting and totally new results: the Brownian Motion.
We refers at the Brownian Motion as a continuous and random motion, usually observable at the scale of some micrometers, of little particles immersed into a fluid. The Brownian Motion was first experienced buy a Dutch physiologist, Jan Ingenhousz, in 1785 and then rediscovered by the Scottish botanic Robert Brown in 1827. Since this latter was the first one who studied the phenomenon in such detail, then it acquired his name. It is curious that even the way in which Brown encountered the first time the experimental evidence of this random, and continuous, motion of little particles is quite random. As a matter of fact, as a botanic, his goal was to study if the random motion of little pollens suspended in water was the manifestation of some kind of life form.
After Brown, many other scientists from different fields dedicated a part of their academical life to the theoretical study of this unpredictable and fascinating phenomenon. Starting in the early 20th century Louis Bachelier, who was among the the firsts to give a theoretical description of the Brownian Motion in term of a stochastic process, was suddenly followed by Albert Einstein, who gave a more physical interpretation in one of his famous works from 1905.
\newpage
\section*{Random Walk}
\newpage
\section*{Wiener Process}
\newpage
\section*{Foccker-Planck Equation}
\newpage
%%%%%%%%%%%%%% Introduction Ends %%%%%%%%%%%%%%%%%%%%%
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%%%%%%%% Main Thesis Starts Here %%%%%%%%%
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\mainmatter % roman numbering
\chapter{Brownian Dynamics}
The theory of Brownian motion is perhaps the simplest approximate way to treat the dynamics of non-equilibrium systems. The fundamental equation is called the Langevin equation and contains both frictional forces and random forces. Moreover, the fluctuation-dissipation theorem relates these forces to each other. As seen before, early investigations of this phenomenon were made on pollen grains, dust particles, and various other objects of colloidal size. Later it became clear that the theory of Brownian motion could be applied successfully to many other phenomena, such as, the motion of ions in water or the reorientation of dipolar molecules.
\section{Langevin Equation}
While the motion of a dust particle performing Brownian motion appears to be quite random, it must in principle be describable by the same equations of motion as is any other dynamical system. In classical mechanics, these are Newton's or Hamilton's equations.\\
Considering the one-dimensional motion of a spherical particle with radius $r$, mass $m$, position $x$ and velocity $v$, in a fluid medium with a known viscosity, the Newton's equation of motion for the particle turn out to be
\subsection{The Over-damped Approximation}
\section{Free Diffusion}
\subsection{Einstein Solution}
\section{Diffusion in Half-Space}
\subsection{Smoluchowski Equation}
\subsection{title}
\chapter{Noise Rectification Models}
\section{Ratchet Model}
\section{Driven Anisotropic Diffusion}
\subsection{Particle Sorting}
Starting from the results shown on the previous section we can derive analytically how to sort particles of different sizes according to the synthetic noise applied to the system. This means that once we know the form of the mean velocity along each direction we can compute the angle that let particle of different sizes to spread in different directions. In order to do that we need to dynamically drive the fluctuations of the synthetic noise in different directions, and with different amplitudes. In this sense, we build a succession of time-steps during which the parameters change. Each of these time-steps, which we could simply be defined as a protocol, would be characterized by different parameters: the direction of the synthetic noise in a 3-Dimensional space, identified by two angles ($\theta$ and $\phi$) according to the expression of general spherical coordinates, and the amplitude of the fluctuations, according to the synthetic temperature $T_{Kin}$ applied to the system.\\
Since this, our main ingredients will be the general vector which drives the direction of the anisotropic temperature
\begin{equation}\label{eq:anis_temp}
\bm{e_{\sigma}}=(\cos\phi \cos\theta, \sin\phi \cos\theta, \sin\theta)
\end{equation}
and the existing relation that links the amplitude of anisotropic fluctuation and the synthetic noise, seen as an external kinetic temperature, applied to the system which is
\begin{equation}
\sigma^{2}=2k_{B}\gamma_{0}(T_{Kin}-T)
\end{equation}
where $k_{B}$ is the Boltzmann constant and $\gamma_{0}$ is the Stokes friction coefficient for a reference spherical particle of radius $r=0.5\mu m$. The Stokes friction coefficient is given by the Stokes formula which states that $\gamma_{0}=6\pi r_{0} \nu$ where $\nu=1.002 \frac{fN s}{\mu m^{2}}$ is the dynamical viscosity of water at an approximate temperature of $300K$.\\
According to \cite{??} we know that in such system the mean velocity along different directions in the long time limit would be
\begin{equation}\label{eq:long_time}
\langle\dot{x_{i}}\rangle:=\lim_{t\to\infty} \frac{1}{t}\langle{x_{i}}\rangle=v_{i}-\frac{D_{3i}}{D_{33}}v_{3}
\end{equation}
where the $D_{ij}$ is the diffusion tensor
\begin{equation}
D_{ij}=\frac{\sigma^{2}}{2}(\gamma^{-1}\bm{e_{\sigma}})_{i}(\gamma^{-1}\bm{e_{\sigma}})_{j}+k_{B}T(\gamma^{-1})_{ij}
\end{equation}
and $v_{i}$ is the velocity proportional to the constant external force applied to the particles and on the frictional coefficient as well
\begin{equation}
v_{i}=(\gamma^{-1})_{ij}f_{j}
\end{equation}
Assuming that the particles have a perfectly round shape (this let $\gamma_{ij}=\gamma\mathds{1}$), since the vector $\bm{e_{\sigma}}$ has the general expression according to \ref{eq:anis_temp}, then the resulting diffusion tensor becomes
\begin{equation}
D_{ij}=\frac{\sigma^{2}}{2\gamma^{2}}(\bm{e_{\sigma}})_{i}(\bm{e_{\sigma}})_{j}+k_{B}T\gamma\mathds{1}
\end{equation}
hence
\begin{equation}\label{eq:Diff_tensor}
D = \begin{pmatrix} \frac{k_{B}T}{\gamma}+\frac{\sigma^{2}\cos^{2}\theta\cos^{2}\phi}{2\gamma^{2}} & \frac{\sigma^{2}\cos^{2}\theta\cos\phi\sin\phi}{2\gamma^{2}} & \frac{\sigma^{2}\cos\theta\sin\theta\cos\phi}{2\gamma^{2}}\\
\frac{\sigma^{2}\cos^{2}\theta\cos\phi\sin\phi}{2\gamma^{2}}
& \frac{k_{B}T}{\gamma}+\frac{\sigma^{2}\cos^{2}\theta\sin^{2}\phi}{2\gamma^{2}}& \frac{\sigma^{2}\cos\theta\sin\theta\sin\phi}{2\gamma^{2}}\\
\frac{\sigma^{2}\cos\theta\sin\theta\cos\phi}{2\gamma^{2}} & \frac{\sigma^{2}\cos\theta\sin\theta\sin\phi}{2\gamma^{2}} &\frac{k_{B}T}{\gamma}+\frac{\sigma^{2}\sin^{2}\theta}{2\gamma^{2}}\\
\end{pmatrix}.
\end{equation}
Once we know the form of the diffusion tensor it is easy to derive the dependence of the mean velocity in each direction according to our protocol's parameters in the long time limit. As seen before, according to \ref{eq:long_time} on the $(x_{1}, x_{2})$ plane we derive the following velocities
\begin{equation}
\langle\dot{x_{1}}\rangle=\frac{1}{\gamma}[f_{1}-f_{3}\frac{D_{13}}{D_{33}}]
\end{equation}
\begin{equation}
\langle\dot{x_{2}}\rangle=\frac{1}{\gamma}[f_{2}-f_{3}\frac{D_{23}}{D_{33}}]
\end{equation}
where $\bm{f}=(f_{1},f_{2},f_{3})$ is again the external force applied to the system.
Since this, using the form of the diffusion tensor from Eq. \ref{eq:Diff_tensor} and rearranging the structure with some simple algebraic manipulation we get the following relations
\begin{equation}
\langle\dot{x_{1}}\rangle=\frac{1}{\gamma}[f_{1}-f_{3}\frac{(T_{Kin}-T)\cos\theta\sin\theta\sin\phi}{\frac{\gamma}{\gamma_{0}}T+(T_{Kin}-T)\sin^{2}\theta}]
\end{equation}
\begin{equation}
\langle\dot{x_{2}}\rangle=\frac{1}{\gamma}[f_{2}-f_{3}\frac{(T_{Kin}-T)\cos\theta\sin\theta\cos\phi}{\frac{\gamma}{\gamma_{0}}T+(T_{Kin}-T)\sin^{2}\theta}]
\end{equation}
which, introducing the new variables $\widetilde{T}=\frac{T_{Kin}-T}{T}$ and $\widetilde{\gamma}=\frac{\gamma}{\gamma_{0}}$, turn into
\begin{equation}
\langle\dot{x_{1}}\rangle=\frac{1}{\gamma}[f_{1}-f_{3}\frac{\widetilde{T}\cos\theta\sin\theta\sin\phi}{\widetilde{\gamma}+\widetilde{T}\sin^{2}\theta}]
\end{equation}
\begin{equation}
\langle\dot{x_{2}}\rangle=\frac{1}{\gamma}[f_{2}-f_{3}\frac{\widetilde{T}\cos\theta\sin\theta\cos\phi}{\widetilde{\gamma}+\widetilde{T}\sin^{2}\theta}].
\end{equation}
At this point, is quite easy to outline the dependence of the mean velocities according to the variables that we can change by applying different protocols to the system. This means that by keeping the external force constant during the overall diffusion process and changing each time step the set $(\theta_{i},\phi_{i},T_{Kin_{i}})$, according to the $i-th$ protocol, we can "synthetically" drive different particles in different directions.\\
This idea would be much clear if we look at the problem in more depth by taking into account the ratio between the velocities of different particles in the same direction.
Assume that we have particles of different radius $r_{i}$; this yields a friction coefficient $\gamma_{i}=6\pi\nu r_{i}$ which is different for every size-type particle. If two particles, which radius are labelled with $r_{a}$ and $r_{b}$ than we need to take in consideration 2 different friction coefficients.
Since both the long term velocities are functions of $(\theta,\phi,T_{Kin})$ then both the ratios in the two directions will be functions of such parameters. Seeing that, is in our interest to study how the particles spread in the $(x_{1},x_{2})$ plane, we can assume $\theta_{i}=\theta$ $\forall i$, that means the angle in the $(x_{2},x_{3})$ plane is the same for each protocol, and there are no external forces which push our particles in such plane so that $\bm{f}=(0,0,f_{3})$. According to this, looking at the mean velocity in the $x_{2}$ direction we can compute the ratio between the velocities of the two particles for the $i-th$ protocol as follows
\begin{equation*}
R_{i}(\phi_{i},T_{Kin_{i}})=\frac{\langle{\dot{x_{2_{A}}}(\phi_{i},T_{Kin_{i}})}\rangle}{\langle{\dot{x_{2_{B}}}(\phi_{i},T_{Kin_{i}})}\rangle}=\frac{\gamma_{B}}{\gamma_{A}}[\frac{f_{3}\frac{\widetilde{T}(T_{Kin_{i})}\cos\theta\sin\theta\cos\phi_{i}}{\widetilde{\gamma}_{A}+\widetilde{T}(T_{Kin_{i}})\sin^{2}\theta}]}{{f_{3}\frac{\widetilde{T}(T_{Kin_{i})}\cos\theta\sin\theta\cos\phi_{i}}{\widetilde{\gamma}_{B}+\widetilde{T}(T_{Kin_{i}})\sin^{2}\theta}]}}]
\end{equation*}
which turns out to be
\begin{equation}\label{eq:R0}
R_{i}(T_{Kin_{i}})=\frac{\gamma_{B}}{\gamma_{A}}[\frac{\widetilde{\gamma}_{B}+\widetilde{T}(T_{Kin_{i}})\sin^{2}\theta}{\widetilde{\gamma}_{A}+\widetilde{T}(T_{Kin_{i}})\sin^{2}\theta}].
\end{equation}
Cause to our assumptions, there are two interest aspects to outline from the last relation. First, since $f_{1}=f_{2}=0$, the ratio for each protocol is the same in both directions. Moreover, what is more interesting is that there is no more dependence on the $\phi_{i}$ angle and so the only parameter which contributes to the ratio is the "synthetic" temperature $T_{Kin_{i}}$.\\
For the sake of completeness, it is good to notice how these considerations are no more verified if $f_{2}\neq f_{1}\neq 0$. In this case the ratio would be expressed by the more general
\begin{equation}\label{eq:ratio}
R_{i}(\phi_{i},T_{Kin_{i}})=R_{0}\frac{f_{2}(\widetilde{\gamma}_{A}+\widetilde{T}(T_{Kin_{i}})\sin^{2}\theta)-f_{3}\widetilde{T}(T_{Kin_{i}})\cos\theta\sin\theta\sin\phi_{i}}{f_{2}(\widetilde{\gamma}_{B}+\widetilde{T}(T_{Kin_{i}})\sin^{2}\theta)-f_{3}\widetilde{T}(T_{Kin_{i}})\cos\theta\sin\theta\sin\phi_{i}}
\end{equation}
where $R_{0}=R_{0}(T_{Kin_{i}})$ is the ratio given in Eq. \ref{eq:R0}.
\subsection{Sequence of protocols}
Since we know how the ratio depends on the protocol parameters we can look for a practical case.
Assume that at time $t=0$ we have a mix of particles, of 2 different sizes, in the centre of our $(x_{1},x_{2})$ plane. The little one, labelled by A, will have a radius $r_{A}=0.2\mu m$ with $\gamma_{A}=6\pi\nu r_{A}$ and the big one with $r_{B}=0.8\mu m$ will have $\gamma_{B}=6\pi\nu r_{B}$.\\
We set two different protocols with fixed $\theta$ such that for the first $500s$ of the diffusion process all particles will spread in the $\phi_{1}$ direction with intensity according to $T_{Kin_{1}}$. The ratio between the velocity along the $x_{2}$ direction (same procedure for $x_{1}$) of the two particles in this first step will be, according to Eq. \ref{eq:ratio}, $R_{1}=R(\phi_{1},T_{Kin_{1}})$. The diffusion during the following $500s$ will then be related to $(\phi_{2}, T_{Kin_{2}})$ from the $2nd$ protocol.\\
Looking at the problem from a more geometrical point of view our goal is to quantify the angle between the resulting directions of the two different particle's type, once both the protocols have been applied.
\end{document}