Dissipative vs. conservative

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In summary, phase space is a mathematical concept that represents all possible states of a system. In a conservative system, momentum is conserved and a set of position and momentum statepoints remains constant. In a dissipative system, momentum decreases as energy is lost through motion.
  • #1
broegger
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Hi.

I'm reading a book on deterministic chaos, and an important distinction seems to be the one between dissipative and conservative dynamical systems. A dissipative system is defined as a system whose "phase space volumes shrink", whereas in a conservative system phase space volumes are conserved.

What does this mean?
 
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  • #2
Nice discussion of Phase Space
http://en.wikipedia.org/wiki/Phase_space

In mathematics and physics, phase space is the space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. A plot of position and momentum variables as a function of time is sometimes called a phase diagram.

In a conservative system, momentum (velocity) is conserved, i.e. not dissipated (reduced), and a particular set (domain) of position and momentum (velocity) statepoints remains. In SHM, e.g. pendulum, the velocity and position have a preserved relationship.

In a dissipative system, the momentum (velocity) is continually decreasing as a function of position/displacement, as energy is lost (dissipated) with continued motion, until the magnitude of velocity (speed) reaches zero as kinetic energy reaches zero.
 
  • #3


In simple terms, a dissipative system is one in which the energy or information within the system is lost over time, leading to a decrease in the system's overall complexity. On the other hand, a conservative system is one in which the energy or information is conserved, meaning that the system's complexity remains constant over time.

To better understand this concept, think of a ball rolling down a hill. In a dissipative system, the ball will eventually come to a stop due to external factors like friction and air resistance, losing its kinetic energy and reducing its complexity. In a conservative system, the ball will continue to roll without any external forces acting on it, maintaining its kinetic energy and complexity.

This distinction is important in the study of chaotic systems because it helps us understand how the system's behavior will change over time. In a dissipative system, the system will eventually reach a state of equilibrium or stability, while a conservative system will continue to exhibit complex and unpredictable behavior.

Overall, the difference between dissipative and conservative systems is a fundamental aspect of dynamical systems and plays a crucial role in understanding and predicting their behavior.
 

1. What is the difference between dissipative and conservative systems?

Dissipative systems are those that involve energy loss, while conservative systems are those that conserve energy. In dissipative systems, energy is converted into a form that cannot be recovered, such as heat or sound. In conservative systems, energy is neither created nor destroyed, but rather is transferred between different forms.

2. How do dissipative and conservative systems affect the behavior of a system?

Dissipative systems tend to decrease in complexity and become more chaotic over time, as energy is continuously lost. On the other hand, conservative systems maintain their complexity and are more stable. This can lead to different behaviors and outcomes in the long term.

3. Can a system be both dissipative and conservative?

Yes, some systems can exhibit both dissipative and conservative behavior, depending on the scale at which they are observed. For example, a system may appear to be dissipative on a macroscopic level, but at a microscopic level, it may exhibit conservative behavior.

4. What are some examples of dissipative and conservative systems?

Examples of dissipative systems include a swinging pendulum, where the energy is gradually lost due to friction and air resistance, and a cup of hot coffee that cools down over time. Examples of conservative systems include a planet orbiting the sun, where the total energy of the system remains constant, and a pendulum with no friction, where the energy is conserved.

5. How do dissipative and conservative systems relate to the laws of thermodynamics?

Dissipative systems are governed by the second law of thermodynamics, which states that the total entropy (or disorder) of a system and its surroundings will always increase over time. On the other hand, conservative systems are governed by the first law of thermodynamics, which states that energy cannot be created or destroyed, but can only be transferred or converted between different forms.

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