Is thermal noise a statistical uncertainty?

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In summary, the conversation discussed a system described by ##y=ax##, where a is the parameter to be extracted and y is the measured quantity. The parameter x is experimentally controlled but has associated uncertainty. It was clarified that x represents the position of a particle in contact with a thermal bath at temperature T, with the particle's energy being ##kx^2/2##. The probability of a given x was stated to follow a Boltzmann distribution. It was questioned whether this uncertainty is of a statistical nature and the concept of "Gaussian and not thermal noise" was brought up. The conversation ended with a discussion on the nature of the thermal probability distribution and the meaning of the "true" value.
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kelly0303
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Hello! I have a system described by ##y=ax##, where a is the parameter I want to extract and y is the stuff I measure (we can assume that I can measure one instance of y without any uncertainty). x is a parameter I can control experimentally but it has an uncertainty associated to it. In a simplified form (but enough for my question), x is the position of a particle (classically) in contact with a thermal bath at temperature T. For example we can assume that the energy of the particle is ##kx^2/2##, where k is a known constant and for each measurement of y, x has a different x, where the probability of an x is given by the probability of having that given energy based on a Boltzman distribution at temperature T. I am not sure if this is a statistical uncertainty or not. I would say it is, because if I measure many y values, I can narrow down the true y value (if I assume I have Gaussian and not thermal noise, that would go down as ##1/\sqrt{N}##, where N is the number of measurements, right?), but I wanted to make sure this makes sense.
 
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What do you mean by "Gaussian and not thermal noise"? The thermal probability distribution in your case is proportional to
$$e^{-\beta E}=e^{-\beta kx^2/2}$$
which is also Gaussian. Besides, by the "true" value, do you mean the average value?
 

FAQ: Is thermal noise a statistical uncertainty?

What is thermal noise?

Thermal noise, also known as Johnson-Nyquist noise, is the electronic noise generated by the thermal agitation of charge carriers (usually electrons) inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. This noise is present in all electrical circuits and is a fundamental aspect of thermodynamics.

Is thermal noise considered a statistical uncertainty?

Yes, thermal noise is considered a statistical uncertainty. It arises from the random motion of electrons, which is inherently unpredictable. This randomness means that thermal noise can be described using statistical methods, and its impact on measurements is treated as a form of uncertainty.

How is thermal noise quantified?

Thermal noise is typically quantified using its power spectral density, which is proportional to the temperature and the resistance of the conductor. The noise voltage can be calculated using the formula: \( V_{noise} = \sqrt{4kTRB} \), where \( k \) is Boltzmann's constant, \( T \) is the absolute temperature in Kelvin, \( R \) is the resistance in ohms, and \( B \) is the bandwidth in hertz.

Can thermal noise be reduced?

Thermal noise cannot be completely eliminated, but it can be reduced. Lowering the temperature of the conductor, reducing the resistance, or narrowing the bandwidth of the system are common methods to minimize thermal noise. However, these methods have practical limitations and trade-offs.

What is the impact of thermal noise on measurements?

Thermal noise introduces a fundamental limit to the precision of measurements in electronic systems. It manifests as a random fluctuation that can obscure the true signal, especially in low-level signal applications. Understanding and accounting for thermal noise is crucial in designing sensitive electronic equipment and in interpreting measurement data accurately.

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